8.à$'<=$6,/$1,$75$1=<6725Ï:
:<%Ï5381.7835$&<75$1=<6725$
3U]H]Z\EyUSXQNWXSUDF\HOHPHQWXDNW\ZQHJRQDOH*\UR]XPLHü
Z\EyUSXQNWXQDMHJRFKDUDNWHU\VW\FHSUGRZRQDSL FLRZHMOXEZSROX
URG]LQ\FKDUDNWHU\VW\NQSNROHNWRURZ\FKOXEGUHQRZ\FK
-DN Z\QLND ] RSLVX ZLHONRV\JQDáRZ\FK PRGHOL WUDQ]\VWRUyZ
ELSRODUQ\FK L XQLSRODUQ\FK RSLVDQ\FK RGSRZLHGQLR Z UR]G] RUD]
GODMHGQR]QDF]QHJRRNUHOHQLDSXQNWXSUDF\WUDQ]\VWRUDNRQLHF]QH
MHVW SRGDQLH GZyFK ZLHONRFL VSRUyG QDSL ü L SUGyZ RSLVXMF\FK
URG]LQ\ MHJR FKDUDNWHU\VW\N VWDW\F]Q\FK : SU]\SDGNX WUDQ]\VWRUD
ELSRODUQHJR ZLHONRFLDPL W\PL V QDMF] FLHM SUG NROHNWRUD
I
C
LQDSL FLH NROHNWRU HPLWHU
U
CE
OXE QDSL FLH ED]D HPLWHU
U
BE
:SU]\SDGNXWUDQ]\VWRUDXQLSRODUQHJRVWRSUGGUHQX
I
D
LQDSL FLH
GUHQ(UyGáR
U
DS
OXEQDSL FLHEUDPND(UyGáR
U
GS
'REyU SXQNWX SUDF\ Z ]QDF]Q\P VWRSQLX ]DOH*\ RG SU]H]QDF]HQLD
LZDUXQNyZSUDF\XNáDGX3RGXZDJ QDOH*\Z]LüF]\QQLNL]ZL]DQH
]HVWUXNWXUXNáDGXVWDZLDQ\PLSU]HGQLPZ\PDJDQLDPLWHFKQLF]Q\PL
RUD]URG]DMHPX*\W\FKWUDQ]\VWRUyZ
3XQNWSUDF\SRZLQLHQOH*HüZREV]DU]HZNWyU\PWUDQ]\VWRU\Z\ND]XM
OLQLRZHZáDFLZRFLZ]PDFQLDMFH'ODWUDQ]\VWRUyZELSRODUQ\FKMHVWWR
REV]DU SUDF\ DNW\ZQHM SU]\ SRODU\]DFML QRUPDOQHM GOD WUDQ]\VWRUyZ
XQLSRODUQ\FKREV]DUQDV\FHQLD
8
&( PD[
X
&(
0
L
&
,
& PD[
3
& PD[
,
& PLQ
8
&( PLQ
8
'6 PD[
X
'6
0
L
'
,
' PD[
3
'PD[
,
'PLQ
8
8
8
'6
*6
7
=
−
b)
a)
5\V 'RSXV]F]DOQH REV]DU\ Z\ERUX SXQNWX SUDF\ Z SROX FKDUDNWHU\VW\N
Z\MFLRZ\FKDWUDQ]\VWRUDELSRODUQHJREWUDQ]\VWRUDXQLSRODUQHJR
'OD ND*GHJR WUDQ]\VWRUD PR*QD Z\]QDF]\ü Z SROX MHJR
FKDUDNWHU\VW\NZ\MFLRZ\FKGRSXV]F]DOQ\REV]DUSUDF\SR]DNWyU\QLH
SRZLQLHQ Z\FKRG]Lü FKZLORZ\ SXQNW SUDF\ Z\]QDF]RQ\ FKZLORZ\PL
ZDUWRFLDPLSUGXLQDSL FLD
-DN SRND]DQR QD U\VD REV]DU WHQ GOD WUDQ]\VWRUD ELSRODUQHJR
RJUDQLF]DMOLQLH
−
PLQLPDOQHJR SUGX NROHNWRUD
I
C min
SRQL*HM NWyUHJR WUDQ]\VWRU
ZFKRG]L Z VWDQ RGFL FLD Z\VWSL ZWHG\ GX*H ]QLHNV]WDáFHQLD
QLHOLQLRZH
−
PDNV\PDOQHJRSUGXNROHNWRUD
I
C max
SRZ\*HMNWyUHJRSRMDZLDMVL
]QLHNV]WDáFHQLD]ZL]DQH]H]PQLHMV]DQLHPZ]PRFQLHQLDSUGRZHJR
β
SU]\GX*\FKSUGDFK
−
PLQLPDOQHJR QDSL FLD
U
U
CE
CEsat
min
=
ZFKRG]HQLH Z ]DNUHV
QDV\FHQLDZL*HVL ]GX*\PL]QLHNV]WDáFHQLDPLQLHOLQLRZ\PL
−
PDNV\PDOQHJRQDSL FLD
U
CE max
RJUDQLF]HQLHZ\QLNDMFH]H]MDZLVND
SRZLHODQLDODZLQRZHJROXE]MDZLVND=HQHUD
−
PDNV\PDOQHM PRF\ VWUDW
P
C max
Z\QLNDMFHM ] PR*OLZRFL
UR]SURV]HQLD UHGQLHM PRF\ Z\G]LHORQHM Z WUDQ]\VWRU]H
LPDNV\PDOQHMWHPSHUDWXU\VWUXNWXU\WUDQ]\VWRUD
3RGREQ\ REV]DU Z SROX FKDUDNWHU\VW\N Z\MFLRZ\FK WUDQ]\VWRUD
SRORZHJR
i
f u
D
DS
=
(
)
SU]HGVWDZLRQRQDU\VE
2JUDQLF]DMJROLQLH
−
SDUDEROD
U
U
U
DS
GS
T
=
−
UR]JUDQLF]DMFD REV]DU OLQLRZ\ WULRGRZ\
RG REV]DUX QDV\FHQLD W\ONR Z REV]DU]H QDV\FHQLD WUDQ]\VWRU
FKDUDNWHU\]XMHVL OLQLRZ\PLZáDFLZRFLDPLZ]PDFQLDMF\PL
−
PLQLPDOQHJRSUGXGUHQX
I
D min
SRQL*HMNWyUHJR WUDQ]\VWRU ZFKRG]L
Z VWDQ RGFL FLD FR ZL*H VL ] GX*\PL ]QLHNV]WDáFHQLDPL
QLHOLQLRZ\PL
−
PDNV\PDOQHJR SUGX GUHQX
I
D max
SRZ\*HM NWyUHJR Z\VWSL
]QLHNV]WDáFHQLD QLHOLQLRZH VSRZRGRZDQH Uy*Q\P QDFK\OHQLHP
FKDUDNWHU\VW\N Z\MFLRZ\FK Uy*Q NRQGXNWDQFM Z\MFLRZ
WUDQ]\VWRUD
−
PDNV\PDOQHJR QDSL FLD
U
DS max
RJUDQLF]HQLH Z\QLNDMFH ]
SRZLHODQLD ODZLQRZHJR QRQLNyZ Z VáDER GRPLHV]NRZDQ\P
REV]DU]HSRGáR*D
−
PDNV\PDOQHM PRF\ VWUDW
P
D max
Z\QLNDMFHM ] PR*OLZRFL
UR]SURV]HQLDUHGQLHMPRF\Z\G]LHORQHMZWUDQ]\VWRU]H
2GZ\EUDQHJRSXQNWXSUDF\WUDQ]\VWRUD]DOH*SUDZLHZV]\VWNLH
SDUDPHWU\WUDQ]\VWRUDSU]\F]\P]DOH*QRüWDMHVWQLHNLHG\EDUG]RVLOQD
1DMF] FLHM QLH MHVW PR*OLZH X]\VNDQLH GOD Z\EUDQHJR SXQNWX SUDF\
ZV]\VWNLFKSDUDPHWUyZ]PLHQQRSUGRZ\FKRRSW\PDOQ\FKZDUWRFLDFK
GOD ]DSHZQLHQLD VWDZLDQ\FK Z\PDJD SURMHNWRZ\FK WDNLFK MDN
ZLHONRü Z]PRFQLHQLD LPSHGDQFMD ZHMFLRZD L Z\MFLRZD V]XP\
SDVPRSU]HQRV]HQLD]QLHNV]WDáFHQLDQLHOLQLRZHPRFZ\MFLRZDLWS1D
RJyá Z\EyU SXQNWX SUDF\ MHVW NRPSURPLVHP SRPL G]\ UR]ELH*Q\PL
Z\PDJDQLDPL NWyU\ QDMOHSLHM MHVW GRNRQDü LQG\ZLGXDOQLH GOD
SURMHNWRZDQHJRXNáDGX
67$7<&=1( , '<1$0,&=1( 35267( 52%2&=( 8.à$'Ï:
:=0$&1,$-&<&+
1D U\VD SU]HGVWDZLRQR VFKHPDW LGHRZ\ Z]PDFQLDF]D
ZNRQILJXUDFMLZVSyOQHJRHPLWHUDZNWyU\P]DVWRVRZDQRQDMSURVWV]\
]PR*OLZ\FK VSRVREyZ ]DVLODQLD WUDQ]\VWRUD SU]\ Z\NRU]\VWDQLX
MHGQHJR(UyGáD]DVLODMFHJRLUH]\VWRUyZW]ZXNáDG]DVLODQLD]HVWDá\P
SUGHPED]\
a)
b)
X
&(
0
L
&
,
%
,
%4
8
&(4
8
&(4
, 5
&4 &
4
,
%
,
&4
8
&&
−
1
5
&
N a c h y l e n i e
8
5
&&
&
0
L
%
5
&
5
%
X
%(
c)
8
5
&&
%
8
&(
=
8
8
&(
&&
=
4
,
%4
, 5
%4 %
8
%(4
8
&&
L
%
L
&
5
%
5
&
L
J
+
8
&&
X
&(
L
E
L
F
5
%
5
&
L
J
X
FH
5\V 3URVW\ SU]\NáDG XNáDGX ]DVLODQLD WUDQ]\VWRUD ELSRODUQHJR D VFKHPDW
LGHRZ\EVFKHPDW]PLHQQRSUGRZ\FJUDILF]QHZ\]QDF]HQLHSXQNWXSUDF\
'ODXNáDGX]U\VDSU]\RGáF]RQ\PRGZHMFLD]PLHQQRSUGRZ\P
(UyGOHi
g
PR*HP\]DSLVDüQDVW SXMFHUyZQDQLD.LUFKKRIID
U
i R
u
CC
C
C
CE
=
+
U
i R
u
CC
B
B
BE
=
+
SU]\ F]\P Z\VW SXMFH Z SRZ\*V]\FK UyZQDQLDFK ZDUWRFL FKZLORZH
SUGyZLQDSL üR]QDF]DMZDUWRFLVWDáRSUGRZHZSXQNFLHSUDF\
8NáDGUyZQDPR*HP\UR]ZL]DüJUDILF]QLHMDNWRSRND]DQR
QDU\VF
5yZQDQLHMHVWUyZQDQLHPW]Z VWDW\F]QHMSURVWHMURERF]HMZSROX
FKDUDNWHU\VW\N Z\MFLRZ\FK
i
f u
C
CE
=
(
)
WUDQ]\VWRUD SU]HFKRG]FHM
SU]H] SXQNW SUDF\ I
CQ
8
&(4
L SU]HFLQDMFHM R
i
C
Z SXQNFLH
U
R
CC
C
RUD]R
u
CE
ZSXQNFLH
U
CC
:SXQNFLHSUDF\VSHáQLRQDMHVWUHODFMD
U
U
I
R
CC
CEQ
CQ
C
=
+
-H*HOL GR ZHMFLD XNáDGX ]RVWDQLH GRáF]RQH ]PLHQQRSUGRZH (UyGáR
SUGX i
g
WR XNáDG UyZQD .LUFKKRIID RERZL]XMH GOD
FKZLORZ\FKZDUWRFLQDSL üLSUGyZWUDQ]\VWRUD
U
i R
u
CC
C
C
CE
=
+
(
)
U
i
i R
u
CC
B
g
B
BE
=
−
+
SU]\ F]\P FKZLORZH ZDUWRFL QDSL ü L SUGyZ V Z\QLNLHP QDáR*HQLD
VNáDGRZ\FK ]PLHQQ\FK QDSL ü L SUGyZ QD VNáDGRZH VWDáH RNUHODMFH
SXQNWSUDF\WUDQ]\VWRUD
i
I
i
B
BQ
b
=
+
i
I
i
u
U
u
C
CQ
c
CE
CEQ
ce
=
+
=
+
1D U\VE SU]HGVWDZLRQR UyZQRZD*Q\ VFKHPDW ]PLHQQRSUGRZ\
UR]SDWU\ZDQHJR XNáDGX Z NWyU\P GOD VNáDGRZ\FK ]PLHQQ\FK
RERZL]XM]DOH*QRFL
u
i R
ce
c
C
= −
3RGVWDZLDMF]DOH*QRFLGRUyZQDQLDRWU]\PXMHP\
i
R
u
I
U
R
C
C
CE
CQ
CEQ
C
− −
+
+
1
5yZQDQLH SU]HGVWDZLD W]Z G\QDPLF]Q SURVW URERF] Z SROX
FKDUDNWHU\VW\N Z\MFLRZ\FK WUDQ]\VWRUD NWyUD SU]HFKRG]L SU]H] SXQNW
SUDF\ D MHM QDFK\OHQLH
−
1 R
C
RGSRZLDGD Z\SDGNRZHM UH]\VWDQFML Z
REZRG]LH Z\MFLRZ\P WUDQ]\VWRUD GOD VNáDGRZHM ]PLHQQHM
:UR]ZD*DQ\PSU]\SDGNXRELHSURVWHURERF]HVWDW\F]QDLG\QDPLF]QD
SRNU\ZDMVL
$E\ MHGQDN SRND]Dü Uy*QLF SRPL G]\ VWDW\F]Q D G\QDPLF]Q SURVW
URERF] UR]ZD*P\ QLHFR LQQ\ XNáDG SU]HGVWDZLRQ\ QD U\VD
:XNáDG]LH W\P Z REZRG]LH HPLWHUD MHVW ZáF]RQD UH]\VWDQFMD
R
E
]ERF]QLNRZDQD SRMHPQRFL
C
E
WDN GREUDQD *H GOD F] VWRWOLZRFL
V\JQDáyZ SU]HQRV]RQ\FK SU]H] Z]PDFQLDF] MHM UHDNWDQFMD MHVW
SRPLMDOQLHPDáDNRQGHQVDWRU
C
E
VWDQRZL]ZDUFLH
a)
b)
X
&(
0
L
&
,
%1
,
%4
8
&(4
8
&(4
, 5
&4 &
4
,
%
,
&4
8
&&
−
1
5
&
Nachylenie
8
5
&&
&
, 5
&4 (
8
5
5
&&
&
(
+
(
)
−
+
1
5 5
&
(
Nachylenie
c)
L
%
L
&
5
%
5
&
L
J
+
8
&&
X
&(
5
(
&
(
L
E
L
F
5
%
5
&
L
J
X
FH
5\V8NáDG]DVLODQLDWUDQ]\VWRUDELSRODUQHJR]GZyMQLNLHP 5 &
(
(
__
ZREZRG]LH
HPLWHUDDVFKHPDWLGHRZ\EVFKHPDW]PLHQQRSUGRZ\FVWDW\F]QDLG\QDPLF]QD
SURVWDSUDF\ZSROXFKDUDNWHU\VW\NZ\MFLRZ\FK
: W\P SU]\SDGNX UyZQDQLH .LUFKKRIID GOD REZRGX NROHNWRUD SU]\
RGáF]RQ\P]PLHQQRSUGRZ\P(UyGOHi
g
RSLVXMH]DOH*QRü
(
)
U
i R
R
u
CC
C
C
E
CE
=
+
+
SU]\F]\P
i
C
u
CE
VZDUWRFLDPLVWDáRSUGRZ\PLRNUHODMF\PLSXQNW
SUDF\
5yZQDQLH RSLVXMH VWDW\F]Q SURVW URERF] SU]HFKRG]F SU]H]
SXQNWSUDF\ I
CQ
U
CEQ
LSU]HFLQDMFHMR
i
C
ZSXQNFLH
(
)
U
R
R
CC
C
E
+
RUD]R
u
CE
ZSXQNFLH
U
CC
U\VF
'OD VNáDGRZ\FK ]PLHQQ\FK UH]\VWDQFMD
R
E
MHVW ]ZDUWD ]DWHP
]PLHQQRSUGRZ\ VFKHPDW XNáDGX U\VE MHVW LGHQW\F]Q\ MDN
ZSRSU]HGQLR UR]ZD*DQ\P SU]\NáDG]LH =DWHP G\QDPLF]Q SURVW
URERF]RSLVXMHZW\PSU]\SDGNXUyZQLH*UyZQDQLH-DNSRND]DQR
QD U\VF RELH SURVWH URERF]H SU]HFKRG] SU]H] SXQNW SUDF\
RZVSyáU] GQ\FK I
CQ
L U
CEQ
SU]\ F]\P LFK QDFK\OHQLD Z\QRV]
RGSRZLHGQLRVWDW\F]QHM
(
)
−
+
1 R
R
C
E
]DG\QDPLF]QHM
−
1 R
C
-DNRRVWDWQLSU]\NáDGUR]ZD*\P\XNáDGZ]PDFQLDMF\ZNWyU\P
REFL*HQLH GRáF]RQH MHVW GR REZRGX NROHNWRURZHJR WUDQ]\VWRUD ]D
SRUHGQLFWZHPWUDQVIRUPDWRUDVSU] JDMFHJRU\V
a)
4
nachylenie
b)
8
8
&(4
&&
=
2
8
&&
X
&(
L
&
Prosta statyczna
∞
nachylenie
Prosta dynamiczna
−
5
/
,
&4
+
8
&&
X
L
5
5
/
5
S 5
/
/
=
2
S 1
X
/
5\V:]PDFQLDF]]HVSU] *HQLHPWUDQVIRUPDWRURZ\PDVFKHPDWLGHRZ\
ESURVWHURERF]HVWDW\F]QDLG\QDPLF]QDZXNáDG]LH
-H*HOLUH]\VWDQFMDX]ZRMHQLDSLHUZRWQHJRWUDQVIRUPDWRUDMHVWSRPLMDOQLH
PDáDWRVWDW\F]QDSURVWDURERF]DSU]HFKRG]LSLRQRZRSU]H]SXQNWSUDF\
Q
SU]HFLQDMF R
u
CE
Z SXQNFLH U
U
CEQ
CC
=
'\QDPLF]QD SURVWD
URERF]D SU]HFKRG]L SU]H] SXQNW SUDF\ Q D MHM QDFK\OHQLH
−
1 R
L
'
RGSRZLDGDSU]HWUDQVIRUPRZDQHMUH]\VWDQFMLREFL*HQLD R
p R
L
L
'
=
2
SU]\
F]\P pMHVWSU]HNáDGQLWUDQVIRUPDWRUD
8VWDOHQLH Z\EUDQHJR SXQNWX SUDF\ L ]DSHZQLHQLH MHJR VWDáRFL
SU]\]PLDQDFKF]\QQLNyZ]HZQ WU]Q\FKWDNLFKMDN]PLDQ\WHPSHUDWXU\
QDSL ü]DVLODMF\FKF]\Z\PLDQLHHOHPHQWyZZ\PDJD]DSURMHNWRZDQLD
RGSRZLHGQLHJR XNáDGX ]DVLODQLD L VWDELOL]DFML SXQNWyZ SUDF\
WUDQ]\VWRUyZZSURMHNWRZDQ\PXNáDG]LH
=$6,/$1,(
,
67$%,/,=$&-$
381.7Ï:
35$&<
75$1=<6725Ï::8.à$'$&+'<6.5(71<&+
8NáDG\]DVLODQLDWUDQ]\VWRUyZELSRODUQ\FK
'R DQDOL]\ VWDáRSUGRZ\FK ZDUXQNyZ SUDF\ Z REV]DU]H
DNW\ZQ\PZ\NRU]\VWDP\RPyZLRQ\ZUR]G]PRGHO(EHUVD0ROOD
U\V
B
,
&
C
E
,
(
α
1 (
,
,
%
8
%(
B
,
&
C
E
,
(
α
1 (
,
,
%
8
%(
,
&0
B
,
&
C
E
,
(
β
1 %
,
,
%
8
%(
(
)
,
,
&(
1
&
0
0
1
=
+
β
5\V0RGHOVWDáRSUGRZ\WUDQ]\VWRUDELSRODUQHJR
=JRGQLH]UR]Sá\ZHPSUGyZZWUDQ]\VWRU]HRWU]\PXMHP\
I
I
I
E
N
E
C
=
+
α
0
I
I
I
E
N
B
CE
=
+
β
0
(
)
I
I
CE
N
C
0
0
1
=
+
β
I
C 0
SUG]HURZ\NROHNWRUDZNRQILJXUDFML2%
=DWHPR]QDF]DMF
β
β
N
=
0
SUGNROHNWRUD
I
C
Z\QRVL
(
)
I
I
I
C
B
C
=
+
+
β
β
0
0
0
1
7HUPLF]QD ]DOH*QRü NRQFHQWUDFML VDPRLVWQ\FK QRQLNyZ SUGX
UXFKOLZRFL QRQLNyZ SUGX ZVSyáF]\QQLND G\IX]ML F]DVX *\FLD
QRQLNyZRUD]SRWHQFMDáX)HUPLHJRSRZRGXM*HSDUDPHWU\WUDQ]\VWRUD
]DOH*RGWHPSHUDWXU\
0R*QDSU]\Mü*HWHUPLF]QH]PLDQ\SUGXNROHNWRUDZL*VL JáyZQLH
]H ]PLDQDPL WU]HFK ZLHONRFL
I
C 0
U
BE
L
β
0
NWyUH ]DOH* RG
WHPSHUDWXU\ L NWyUH Z VSRVyE MDZQ\ GHWHUPLQXM ZLHONRü SUGX
NROHNWRUDZXNáDG]LH]DVLODQLDWUDQ]\VWRUD
(
)
I
f U
I
C
BE
C
=
,
,
β
0
0
3UG
I
C 0
WUDQ]\VWRUyZ NU]HPRZ\FK MHVW EDUG]R PDá\ Z WHPSHUDWXU]H
°
& U] GX Q$ L Z SU]\EOL*HQLX SRGZDMD VL SU]\ Z]URFLH
WHPSHUDWXU\RND*GH
°
&ZRWRF]HQLXWHPSHUDWXU\
°
&
1DSL FLH
U
BE
Z\QRVL RN 9 Z WHPSHUDWXU]H
°
& L PDOHMH ]H
Z]URVWHPWHPSHUDWXU\RNP9
°
&
:VSyáF]\QQLN
β
0
URQLH ]H Z]URVWHP WHPSHUDWXU\ L Z RSDUFLX
RHPSLU\F]QH ]DOH*QRFL PR*QD VWZLHUG]Lü OLQLRZ\ SU]\URVW
β
0
]V]\ENRFL
°
&
7HUPLF]QH ]PLDQ\ Z\V]F]HJyOQLRQ\FK SDUDPHWUyZ WUDQ]\VWRUD
PRJE\üRSLVDQHSU]\EOL*RQ\PL]DOH*QRFLDPLDQDOLW\F]Q\PL
( )
( )
I
T
I
C
C
C
T
C
0
0
8
25
2
≈
⋅
$
$
∆
( )
( )
U
T
U
C
T
V
C
BE
BE
≈
−
25
0 0022
$
$
,
∆
( )
( )
β
β
0
0
25
1
80
T
C
T
C
≈
⋅ +
$
$
∆
=PLDQ\
β
0
U
BE
I
C 0
PRJZ\QLNDüQLHW\ONR]H]PLDQ\WHPSHUDWXU\
DOH UyZQLH* ] UR]U]XWX SURGXNF\MQHJR SDUDPHWUyZ 6]F]HJyOQLH GX *\
UR]U]XWSURGXNF\MQ\PR*HSRVLDGDüZVSyáF]\QQLN
β
0
1SZNDWDORJDFK
G\VNUHWQH WUDQ]\VWRU\ WHJR VDPHJR W\SX V QDMF] FLHM G]LHORQH QD
JUXS\]DOH*QLHRGZLHONRFLZ]PRFQLHQLDSUGRZHJR
β
0
]HZ]JO GX
QDEDUG]RGX*\UR]U]XWWHFKQRORJLF]Q\WHJRSDUDPHWUX]DZLHUDMF\VL
QDZHWZJUDQLFDFK
3U]\URVW SUGX NROHNWRUD Z\ZRáDQ\ SU]\URVWDPL SRV]F]HJyOQ\FK
ZLHONRFL
β
0
U
BE
I
C 0
PR*HP\ Z\]QDF]\ü Z SRVWDFL Uy*QLF]NL
]XSHáQHMSUGX
I
C
RNUHORQHJRZ]RUHP
dI
I
d
I
U
dU
I
I
dI
C
C
C
BE
BE
C
C
C
≈
+
+
∂
∂ β
β
∂
∂
∂
∂
0
0
0
0
3RFKRGQH F]VWNRZH Z UyZQDQLX V IXQNFMDPL XNáDGRZ\PL
REZRGyZ ]DVLODQLD WUDQ]\VWRUyZ L RNUHOD VL MH PLDQHP
ZVSyáF]\QQLNyZVWDELOL]DFMLSUGXNROHNWRUDZQDVW SXMF\VSRVyE
S
I
I
I
I
I
C
C
C
C
=
≈
∂
∂
0
0
∆
∆
S
I
U
I
U
U
C
BE
C
BE
=
≈
∂
∂
∆
∆
S
I
I
C
C
β
∂
∂ β
=
≈
0
0
∆
∆β
=DWHPUyZQDQLHPR*HP\]DSLVDüZSRVWDFL
∆
∆
∆
∆β
I
S
I
S
U
S
C
I
C
U
BE
≈
+
+
0
0
β
'OD WUDQ]\VWRUyZ NU]HPRZ\FK F] VWR SRPLMD VL ZVSyáF]\QQLN
VWDELOL]DFML
S
I
]HZ]JO GXQDEDUG]RPDáZDUWRüSUGX]HURZHJR
I
C 0
WUDQ]\VWRUDLSRPLMDOQZDUWRüVNáDGQLND
6 ,
,
&
∆
8NáDG]DVLODQLDWUDQ]\VWRUDMHVWW\POHSV]\LPPQLHMV]H]DSHZQLD
ZVSyáF]\QQLNLVWDELOL]DFMLSUGXNROHNWRUDZWHG\ERZLHPQDZHWGX*H
]PLDQ\
∆
I
C 0
∆
U
BE
∆β
0
RGSRZLDGDMFH GX*\P ]PLDQRP
WHPSHUDWXU\Z\ZRáXMQLHZLHONLSU]\URVWSUGXNROHNWRUD
-HGQ\P ] QDMSURVWV]\FK XNáDGyZ ]DVLODQLD WUDQ]\VWRUD MHVW
UR]ZD*DQ\ ZF]HQLHM XNáDG ]DVLODQLD VWDá\P SUGHP ED]\ NWyUHJR
VFKHPDWLGHRZ\SRZWyU]RQRQDU\V
,
&
,
(
β
0
,
%
,
%
8
%(
,
&( 0
8
&&
5
&
5
%
8
&&
8
&(
5
&
5
%
,
%
,
&
8
&(
5\V8NáDG]DVLODQLDVWDá\PSUGHPED]\LMHJRPRGHOGODVNáDGRZHMVWDáHM
=HVFKHPDWXQDU\VRWU]\PXMHP\
,
8
8
5
8
5
FRQVW
8
8
%
&&
%(
%
&&
%
&&
%(
=
−
≈
≈
>>
GOD
=]DOH*QRFLZ\QLNDQD]ZDXNáDGX
:\NRU]\VWXMF ]DOH*QRFL RUD] UyZQDQLH .LUFKKRIID GOD
REZRGX NROHNWRURZHJR XNáDGX ] U\V RWU]\PXMHP\ ZVSyáU] GQH
SXQNWXSUDF\WUDQ]\VWRUD
(
)
I
U
U
R
I
U
R
U
U
I R
C
CC
BE
B
C
CC
B
CE
CC
C
C
=
−
+
+
≈
=
−
β
β
β
0
0
0
0
1
: XNáDG]LH W\P QLH PD *DGQ\FK PR*OLZRFL RSW\PDOL]DFML
ZDUWRFLHOHPHQWyZXNáDGXSRODU\]DFMLJG\*GOD]DGDQHJRSXQNWXSUDF\
I
C
U
CE
L QDSL FLD ]DVLODQLD
U
CC
Z\QLNDM MHGQR]QDF]QLH RNUHORQH
ZDUWRFL HOHPHQWyZ
R
B
R
C
NWyUH V NRQVHNZHQFM UyZQD
RNUHODMF\FKSXQNWSUDF\
:DUWRFL ZVSyáF]\QQLNyZ VWDELOL]DFML Z W\P XNáDG]LH V GX*H ]DWHP
SXQNWSUDF\MHVWVLOQLHX]DOH*QLRQ\RGSDUDPHWUyZWUDQ]\VWRUD
/HSV]H ZáDVQRFL VWDELOL]DF\MQH SRVLDGD XNáDG ]DVLODQLD VWDá\P
SUGHPHPLWHUDU\V
,
&
,
(
α
0
,
(
,
%
8
%(
,
&0
8
&&
5
&
5
(
8
&&
8
&(
5
&
5
(
,
%
,
&
8
&(
,
(
−
8
((
C
E
B
−
8
((
5\V8NáDG]DVLODQLDVWDá\PSUGHPHPLWHUDLMHJRPRGHOGODVNáDGRZHMVWDáHM
=HVFKHPDWXSU]HGVWDZLRQHJRQDU\VRWU]\PXMHP\
I
U
U
R
U
R
const
U
U
E
EE
BE
E
EE
E
EE
BE
=
−
≈
≈
>>
(
)
GOD
=]DOH*QRFLZ\QLNDQD]ZDXNáDGX
8Z]JO GQLDMF
I
I
I
C
E
C
=
+
+
β
β
0
0
0
1
]UyZQDLRWU]\PXMHP\
I
U
U
R
I
U
U
R
U
R
C
EE
BE
E
C
EE
BE
E
EE
E
=
+
−
+
≈
−
≈
β
β
0
0
0
1
:SRGREQ\VSRVyEPR*QDREOLF]\üQDSL FLH
U
CE
U
U
I R
U
I R
CC
EE
C
C
CE
E
E
+
=
+
+
=UyZQDRWU]\PXMHP\
(
)
U
U
U
I
R
R
I R
U
U
I
R
R
CE
CC
EE
C
C
E
C
E
CC
EE
C
C
E
=
+
−
+
+
+
+
≈
≈
+
−
+
β
β
β
β
0
0
0
0
0
1
1
-HVW WR UyZQDQLH VWDW\F]QHM SURVWHM SUDF\ Z SROX FKDUDNWHU\VW\N
Z\MFLRZ\FK WUDQ]\VWRUD 5yZQDQLD RNUHODM
ZVSyáU] GQHSXQNWXSUDF\LPR*QD]QLFKMHGQR]QDF]QLHZ\]QDF]\ü
R
C
L
R
E
SU]\ ]DGDQ\FK
U
CC
U
EE
:V]\VWNLH ZVSyáF]\QQLNL VWDELOL]DFML
6 6 6
,
8
β
NWyUH PR*QD Z\]QDF]\ü ] UyZQDQLD REOLF]DMF
SRFKRGQHF]VWNRZHNROHMQRZ]JO GHP
I
C 0
U
BE
L
β
0
PDMPQLHMV]H
ZDUWRFL QL* Z SU]\SDGNX XNáDGX ]D VWDá\P SUGHP ED]\ 6WDELOL]DFMD
SXQNWXSUDF\MHVWW\POHSV]DLP
5
(
MHVWZL NV]H
8NáDG ]DVLODQLD ]H VSU] *HQLHP NROHNWRURZ\P ]RVWDá
SU]HGVWDZLRQ\ QD U\V 6SU] *HQLH ]ZURWQH QDSL FLRZH UHDOL]RZDQH
SU]H]UH]\VWRU
R
F
VWDELOL]XMH]PLDQ\QDSL FLD
U
CE
-H*HOL]PDOHMHSUG
NROHNWRUD
I
C
WR ]PDOHMH VSDGHN QDSL FLD QD UH]\VWRU]H
R
C
UyZQ\
(
)
I
I
R
C
B
C
+
Z]URQLH QDSL FLH
U
CE
]DWHP Z]URQLH QDSL FLH QD
UH]\VWRU]H
R
F
I R
U
U
B
F
CE
BE
=
−
Z]URQLH SUG ED]\
I
B
Z]URQLH
SUGNROHNWRUD
I
C
FR]ZURWQLHVSRZRGXMH]ZL NV]HQLHVSDGNXQDSL FLD
QD UH]\VWRU]H
R
C
F]\OL ]PQLHMV]HQLH QDSL FLD
U
CE
1D W\P SROHJD
VWDELOL]XMFD UROD VSU] *HQLD ]ZURWQHJR : UH]XOWDFLH
U
CE
Z]URQLH
PQLHM QL* Z XNáDG]LH EH] VSU] *HQLD 6SU] *HQLH ]ZURWQH MHVW W\P
VLOQLHMV]H L VWDELOL]DFMD SXQNWX SUDF\ OHSV]D LP ZDUWR ü
R
F
MHVW
PQLHMV]D
,
&
β
0
,
%
,
%
8
%(
,
&( 0
=
8
&&
5
&
5
)
8
&&
8
&(
5
&
5
)
,
%
,
&
8
&(
C
E
B
(
)
=
+
β
0
0
1
,
&
5\V8NáDG]HVSU] *HQLHPNROHNWRURZ\PLMHJRPRGHOGODVNáDGRZHMVWDáHM
'ODXNáDGX]U\VRWU]\PXMHP\
(
)
R
I
I
I
R I
U
U
C
B
B
CE
F
B
BE
CC
β
0
0
+
+
+
+
=
(
)
I
I
I
B
C
C
=
−
+
β
β
0
0
0
1
6WG
(
)
(
)(
)
(
)
I
U
U
I
R
R
R
R
C
CC
BE
C
C
F
F
C
=
−
+
+
+
+
+
β
β
β
0
0
0
0
1
1
(
)
U
I
I
R
U
CE
C
B
C
CC
+
+
=
.RU]\VWDMF ] ]DOH*QRFL L RWU]\PXMHP\ UyZQDQLH
VWDW\F]QHMSURVWHMREFL*HQLDZSROXFKDUDNWHU\VW\NZ\MFLRZ\FK
(
)
U
I
I
R
U
CE
C
C
C
CC
+
+
+
=
β
β
0
0
0
1
=DOH*QRFL L RNUHODM ZVSyáU] GQH SXQNWX SUDF\
WUDQ]\VWRUD
,QQ\ VSRVyE Z\NRU]\VWDQLD XMHPQHJR VSU] *HQLD ]ZURWQHJR GR
VWDELOL]DFML SXQNWX SUDF\ SU]HGVWDZLRQR QD U\V Z XNáDG]LH
SRWHQFMRPHWU\F]Q\P ]H VSU] *HQLHP HPLWHURZ\P -HVW WR SRGVWDZRZ\
VSRVyE ]DVLODQLD WUDQ]\VWRUD Z XNáDGDFK G\VNUHWQ\FK ]DZLHUDM F\FK
MHGQR(UyGáRQDSL FLD]DVLODMFHJR
8
&&
5
&
5
(
,
%
,
&
8
&(
,
(
5
5
5
5 5
%
=
__
tw. Thevenina
8
8
5
5 5
8
5
5
%%
&&
&&
%
=
+
=
8
&&
5
&
5
(
,
%
,
&
8
&(
,
(
5
%
8
%(
8
%%
a)
b)
5\V 3RWHQFMRPHWU\F]Q\ XNáDG ]D VSU] *HQLHP HPLWHURZ\P D L MHJR
UyZQRZD*Q\XNáDG]]DVW SF]\P(UyGáHP7KHYHQLQDE
.RU]\VWDMF ] ]DVDG\ 7KHYHQLQD XNáDG WHQ PR*HP\ SU]HNV]WDáFLü Z
UyZQRZD*Q\ XNáDG ]H VSU] *HQLHP HPLWHURZ\P L ] GZRPD (UyGáDPL
]DVLODQLDU\VE
6SU] *HQLH ]ZURWQH SUGRZH UHDOL]RZDQH SU]H]
R
E
VWDELOL]XMH
]PLDQ\ SUGX
I
E
-H*HOL ]PDOHMH SUG HPLWHUD
I
E
WR ]PDOHMH VSDGHN
QDSL FLDQDUH]\VWRU]H
R
E
]DWHPZ]URQLHQDSL FLH
U
BE
Z]URQLHSUG
ED]\
I
B
FR]ZURWQLHVSRZRGXMHZ]URVWSUGXHPLWHUD
I
E
:UH]XOWDFLH
SUGHPLWHUD
I
E
]PDOHMHPQLHMQL*ZXNáDG]LHEH]VSU] *HQLD]ZURWQHJR
'ODXNáDGX]U\VEPR*HP\]DSLVDüUyZQDQLD
(
)
U
I R
U
I R
I
I
I
I
I
I
BB
B
B
BE
E
E
C
B
C
E
B
C
=
+
+
=
+
+
=
+
β
β
0
0
0
1
]NWyU\FKPR*HP\Z\]QDF]\üSUGNROHNWRUD
(
)
(
)(
)
(
)
I
U
U
I
R
R
R
R
C
BB
BE
C
B
E
B
E
=
−
+
+
+
+
+
β
β
β
0
0
0
0
1
1
:\NRU]\VWXMF UyZQDQLD RUD] UyZQDQLH .LUFKKRIID GOD REZRGX
NROHNWRURZHJR
U
I R
U
I R
CC
C
C
CE
E
E
=
+
+
RWU]\PXMHP\
U
U
I R
I
R
R
CE
CC
C
E
C
C
E
=
+
+
−
+
+
0
0
0
0
0
1
1
β
β
β
β
-HVW WR UyZQDQLH VWDW\F]QHM SURVWHM REFL*HQLD Z SROX Z\MFLRZ\FK
FKDUDNWHU\VW\N
( )
I U
C
CE
.RU]\VWDMF]]DOH*QRFLLáDWZRPR*QDZ\ND]Dü *H
VWDELOL]DFMDSRSUDZLDVL JG\VWRVXQHN
R
R
B
E
→
0
JG\VWDELOL]RZDQ\
MHVW SUG HPLWHUD -HGQDN JG\ ZDUWRü
R
E
MHVW ]E\W GX*D WR QDSL FLH
]DVLODQLD PXVL E\ü GX*H RUD] SRJDUV]D VL VWDELOL]DFMD QDSL FLD
U
CE
DOERSU]\W\PVDP\PVWRSQLXVWDELOL]DFMLQDSL FLD
U
CE
UH]\VWDQFMD
R
C
PXVLE\üPDáDVWUDW\Z]PRFQLHQLD
3U]\MPXMF
I
C 0
0
=
β
β
0
0
1
+ ≈
L
U
const
CC
=
PR*HP\QDSLVDü
(
)
U
U
I
R
R
CE
CC
C
C
E
=
−
+
(
)
−
=
+
∆
∆
U
R
R
I
CE
C
E
C
%H]SLHF]QLHMMHVW]DPLHQLüUyZQDQLHQDQLHUyZQR ü
(
)
−
≥
+
∆
∆
U
R
R
I
CE
C
E
C
R
R
U
I
C
E
CE
C
+
≤ ∆
∆
=E\W PDáD ZDUWRü UH]\VWDQFML
R
B
RJUDQLF]D Z]PRFQLHQLH GOD
VNáDGRZHM]PLHQQHM]HZ]JO GXQDVLOQHWáXPLHQLHZHMFLDWUDQ]\VWRUD
:Sá\Z UH]\VWDQFML
R
E
QD SDUDPHWU\ URERF]H Z]PDFQLDF]D
Z]PRFQLHQLH QDSL FLRZH L LPSHGDQFM ZHMFLRZ GOD VNáDGRZHM
]PLHQQHM PR*QD ]QDF]QLH ]PQLHMV]\ü SRSU]H] ]DVWRVRZDQLH
NRQGHQVDWRUD EORNXMFHJR 7DN ZL F SU]\ Z\ERU]H ZDUWRFL
R
E
R
B
QDOH*\ VL NLHURZDü NRPSURPLVHP SRPL G]\ ZDUXQNDPL VWDELOL]DFML
SUGX
I
C
QDSL FLD
U
CE
ZLHONRFL QDSL FLD ]DVLODMFHJR
U
CC
LZ]PRFQLHQLHPXNáDGX
1DMOHSV]H ZáDVQRFL VWDELOL]DF\MQH PD XNáDG ] GZRPD (UyGáDPL
]DVLODQLD Z W\P XNáDG ]H VWDá\P SUGHP HPLWHUD 0RF SRELHUDQD ]H
(UyGHá ]DVLODQLD Z W\P SU]\SDGNX MHVW ] UHJXá\ PQLHMV]D QL *
ZSU]\SDGNX XNáDGX ] MHGQ\P (UyGáHP 1DWRPLDVW ZDG MHVW
NRQLHF]QRüVWRVRZDQLDGZyFK(UyGHá]DVLODQLD
=DGDZDODMFH HIHNW\ GDMH ]DVWRVRZDQLH XNáDGX ]H VSU] *HQLHP
HPLWHURZ\PNROHNWRURZ\PL]RE\GZRPDQDUD]-HGQDN *HZXNáDG]LH
]H VSU] *HQLHP NROHNWRURZ\P VWDELOL]DFMD ]DOH*\ RG UH]\VWDQFML
R
C
]DWHP XNáDG WHQ MHVW WUXGQLHMV]\ GR UHDOL]DFML =H Z]JO GyZ
VWDELOL]DF\MQ\FKQDMJRUV]HZáDVQRFLPDXNáDG]HVWDá\PSUGHPED]\
8NáDG ]DVLODQLD VWDá\P SUGHP HPLWHUD FKDUDNWHU\]XMF\ VL
QDMNRU]\VWQLHMV]\PL ZáDFLZRFLDPL VWDELOL]DF\MQ\PL QDMáDWZLHM MHVW
]UHDOL]RZDüZSRVWDFL(UyGáDSUGXVWDáHJRZREZRG]LHHPLWHUD
3U]\NáDG UHDOL]DFML SURVWHJR (UyGáD VWDáRSUGRZHJR Z WHFKQLFH
HOHPHQWyZ G\VNUHWQ\FK L MHJR VFKHPDW ]DVW SF]\ SU]HGVWDZLRQR QD
U\V
5
(
,
,
2
&
=
,
(
−
8
((
C
8
'
8
=
8
%(
5
5
(
,
,
2
&
=
,
(
−
8
((
J
HF
J
HE
5
(
,
H
E
α
,
H
J
FE
B'
,
F
8
FE
U
EE
≈
U
R
b)
c)
a)
5\V'UyGáRSUGXVWDáHJRD]GLRG=HQHUDE]X]LHPLRQED]
F]PLHQQRSUGRZ\VFKHPDW]DVW SF]\(UyGáD
:\NRU]\VWXMF SUDZR .LUFKKRIID Z XNáDG]LH QD U\VD PR *HP\
Z\]QDF]\üSUGZ\MFLRZ\(UyGáD
I
I
I
U
U
U
R
U
R
O
C
E
Z
D
BE
E
Z
E
=
=
≈
+
−
≈
α
0
: XNáDG]LH Z\VW SXMH Z]DMHPQD NRPSHQVDFMD QDSL FLD
U
BE
]
QDSL FLHP
U
D
GLRG\ ' ZáF]RQHM V]HUHJRZR ] GLRG =HQHUD Z W\P
UyZQLH*WHPSHUDWXURZ\FK]PLDQW\FKQDSL ü
: XNáDG]LH QD U\VE ] X]LHPLRQ ED] GOD SUGX VWDáHJR SUG
(UyGáDZ\QRVL
I
I
I
U
U
R
U
U
R
O
C
E
EE
BE
E
EE
BE
E
=
=
=
−
+
≈
−
α
β
β
0
0
0
1
5y*QLF]NRZD UH]\VWDQFMD Z\MFLRZD (UyGáD Z\]QDF]RQD Z RSDUFLX R
PDáRV\JQDáRZ\VFKHPDW]DVW SF]\QDU\VFZ\QRVL
r
U
I
g
g
g
o
cb
c
cb
ec
ec
≈
≈
+
≈
'
'
1
1
3U]HGVWDZLRQH (UyGáR SUGRZH SRVLDGD GX* UH]\VWDQFM Z\MFLRZ
NWyUHM SU]\EOL*RQD ZDUWRü MHVW WDND MDN WUDQ]\VWRUD Z NRQILJXUDFML
]X]LHPLRQED]
3U]HGVWDZLRQ\ XNáDG (UyGáD SUGRZHJR MHVW GREU\P SU]\NáDGHP
ZVND]XMF\PQDPR*OLZRüZ\NRU]\VWDQLDHOHPHQWyZQLHOLQLRZ\FKGR
NRPSHQVDFML ZSá\ZX WHPSHUDWXURZ\FK ]PLDQ Z\EUDQ\FK SDUDPHWUyZ
WUDQ]\VWRUD
/LQLRZH REZRG\ ]DVLODQLD WM REZRG\ ]DZLHUDM FH Z\áF]QLH OLQLRZH
UH]\VWRU\]DSHZQLDMVWDáRüSXQNWXSUDF\U] GXNLONXNLONXQDVWX
SU]\]PLDQDFKWHPSHUDWXU\ZSU]HG]LDOH
°
&
: XNáDGDFK Z\PDJDMF\FK ]QDF]QLH ZL NV]HM VWDáRFL SUGX U] GX
VWRVXMH VL REZRG\ ]DVLODQLD ] HOHPHQWDPL QLHOLQLRZ\PL
3U]\NáDGHP XNáDGyZ R WDN GX*\FK Z\PDJDQLDFK VWDáRFL SUGX Z
VSRF]\QNRZ\P SXQNFLH SUDF\ PRJ E\ü (UyGáD SUGRZH ]DVLODMFH
VWRSQLH Z]PDFQLDF]D R VSU] *HQLDFK JDOZDQLF]Q\FK QS ZH
Z]PDFQLDF]X RSHUDF\MQ\P SUGRZH (UyGáD UHIHUHQF\MQH LWS
:QLHOLQLRZ\FK REZRGDFK ]DVLODQLD Z\NRU]\VWXMH VL RJyOQ ]DVDG
NRPSHQVDFML ]PLDQ SDUDPHWUyZ VWDELOL]RZDQHJR HOHPHQWX ]PLDQDPL
SDUDPHWUyZLQQHJRHOHPHQWXREDUG]RSRGREQ\FKZáD FLZRFLDFK
1D U\V SU]HGVWDZLRQR XNáDG ] SRWHQFMRPHWU\F]Q\P
]DVLODQLHPED]\LVSU] *HQLHPHPLWHURZ\PRUD]]NRPSHQVDFM]PLDQ
QDSL FLD
U
BE
5
(
,
&
,
,
,
(
&
&
=
+
≈
β
β
−
8
((
8
'
8
%(
5
8
'
5
5
&
,
,
1
n
Q8
'
+
8
&&
,
%
5\V 8NáDG ]DVLODQLD ] GLRGRZ
NRPSHQVDFM]PLDQQDSL FLD8
%(
=DNáDGDMF*HSUGED]\MHVWSRPLMDOQLHPDá\ZSRUyZQDQLXGRSU GX
G]LHOQLND
I
I
1
2
≈
RUD] SU]\MPXMF
I
I
E
C
≈
]H VFKHPDWX QD U\V
RWU]\PXMHP\
nU
I R
U
I R
U
I R
I
U
nU
R
R
D
BE
E
E
BE
C
E
CC
D
+
=
+
≈
+
=
−
+
2
2
2
1
2
=XNáDGXUyZQDPR*HP\Z\]QDF]\ü
I
C
(
)
(
)
I
R U
nU R
R
R U
R R
R
C
CC
D
BE
E
=
+
−
+
+
2
1
1
2
1
2
:DUXQHNSHáQHMNRPSHQVDFML]PLDQQDSL FLD
U
BE
PR*HP\Z\]QDF]\ü]
SU]\UyZQDQLDGR]HUDSRFKRGQHM
dI
dT
C
=
0
F]\OL
(
)
nR
dU
dT
R
R
dU
dT
D
BE
1
1
2
0
−
+
=
-H*HOL GLRGD L WUDQ]\VWRU V Z\NRQDQH Z SRGREQ\P SURFHVLH
WHFKQRORJLF]Q\P GLRGD ]UHDOL]RZDQD MDNR WUDQ]\VWRU Z SRá F]HQLX
GLRGRZ\PLSUDFXMZWHMVDPHMWHPSHUDWXU]HWR
dU
dT
dU
dT
D
BE
≈
LZDUXQHNSHáQHMNRPSHQVDFMLPDSRVWDü
(
)
nR
R
R
1
1
2
0
−
+
=
F]\OL
R
R
n
2
1
1
= −
1DMF] FLHMVWRVXMHVL GZLHGLRG\
n
=
2
LZyZF]DV
R
R
1
2
=
5R]ZD*DQ\ ZF]HQLHM XNáDG (UyGáD SUGX VWDáHJR QD U\VD MHVW
XNáDGHPVWDELOL]DFML]MHGQGLRG
n
=
1
L
R
2
0
=
8NáDG\]DVLODQLDWUDQ]\VWRUyZXQLSRODUQ\FK
7UDQ]\VWRU 026)(7 ] NDQDáHP ]XER*DQ\P PR*H SUDFRZDü
]DUyZQR ]H Z]ERJDFDQLHP MDN L ]H ]XER*DQLHP -HGQDN*H SUDFD ]H
Z]ERJDFDQLHP RGE\ZD VL GOD PDáHJR SU]HG]LDáX ZDUWRFL QDSL FLD
EUDPND (UyGáR
U
GS
'ODWHJR SRGVWDZRZ\P ]DNUHVHP SUDF\ W\FK
WUDQ]\VWRUyZ MHVW SUDFD ]H ]XER*DQLHP 7UDQ]\VWRU\ WH ] NDQDáHP Q
SUDFXMZREV]DU]HQDV\FHQLDSU]\XMHPQ\FKZDUWRFLDFKQDSL FLD
U
GS
L
GRGDWQLFK ZDUWRFLDFK
U
DS
QDWRPLDVW ] NDQDáHP S JG\ SRODUQRü
ZVSRPQLDQ\FKQDSL üMHVW]PLHQLRQDWMJG\QDSL FLH
U
GS
MHVWGRGDWQLH
D
U
DS
MHVWXMHPQH2WU]\PDQLHQDSL üRSRGDQHMSRODUQRFLXPR*OLZLD
XNáDG]GZRPD(UyGáDPL]DVLODQLDU\V
3UGEUDPNL
I
G
WUDQ]\VWRUyZ026)(7MHVWU] GXSLNRDPSHUyZLMHJR
ZSá\Z Z REZRG]LH SRODU\]DFML PR*QD SRPLQü 3U]\ WDNLP ]DáR*HQLX
XNáDG]U\VDRSLVXM]DOH*QRFL
(
)
U
U
I R
U
U
I
R
R
GS
GG
D
S
DD
DS
D
D
S
= −
−
=
+
+
+
8
''
5
'
5
6
,
'
8
'6
>
,
,
6
'
=
−
8
**
,
*
=
5
*
8
*6
<
8
*6
−
8
''
5
'
5
6
,
'
8
'6
<
,
,
6
'
=
+
8
**
,
*
=
5
*
8
*6
>
8
*6
b)
a)
5\V 8NáDG ] GZRPD (UyGáDPL ]DVLODQLD WUDQ]\VWRUD 026 ] NDQDáHP
]XER*DQ\PDW\SXQEW\SXS
1DWRPLDVW XNáDG ] U\VE ]JRGQLH ] ]DVWU]DáNRZDQ\PL NLHUXQNDPL
SUGyZLQDSL üRSLVXMDUyZQDQLD
(
)
U
U
I R
U
U
I
R
R
GS
GG
D
S
DD
DS
D
D
S
=
+
= −
+
+
1DOH*\]D]QDF]\ü*HZREXXNáDGDFKQDU\V]D]QDF]RQRIL]\F]QH
NLHUXQNLSU]HSá\ZXSUGyZGUHQX
I
D
: REX XNáDGDFK ]DVWRVRZDQR Z REZRG]LH (UyGáD UH]\VWRU
R
S
VWDELOL]XMF\ SRGREQLH MDN UH]\VWRU
R
E
Z REZRG]LH HPLWHUD SXQNW
SUDF\WUDQ]\VWRUDRZVSyáU] GQ\FK I
DQ
U
DSQ
3UREOHPVWDáRFLSXQNWX
SUDF\ZL*HVL ]UR]U]XWHPWHFKQRORJLF]Q\PZDUWRFLQDSL FLDRGFL FLD
U
P
L SUGX QDV\FHQLD
I
DSS
GOD WUDQ]\VWRUyZ ] NDQDáHP ]XER*DQ\P
QDSL FLD SURJRZHJR
U
T
L ZVSyáF]\QQLND
K W L
'
GOD WUDQ]\VWRUyZ
]NDQDáHPZ]ERJDFDQ\P RUD] ZSá\ZHP WHPSHUDWXU\ QD WH SDUDPHWU\
7HUPLF]QD]DOH*QRüUXFKOLZRFLQRQLNyZSUGXZNDQDOHRUD]QDSL ü
U
P
L
U
T
VJáyZQ\PF]\QQLNLHPWHPSHUDWXURZ\FK]PLDQSUGXGUHQX
,VWQLHMH MHGQDN WDND ZDUWRü SUGX GUHQX SU]\ NWyUHM ZSá\Z\ WH
NRPSHQVXM VL 3UG WHQ QRVL QD]Z SUGX DXWRNRPSHQVDFML :DG
SU]HGVWDZLRQ\FK QD U\V XNáDGyZ SRODU\]DFML WUDQ]\VWRUD MHVW
NRQLHF]QRü ]DVWRVRZDQLD GZyFK (UyGHá ]DVLODQLD L WR RGPLHQQHM
SRODUQRFL
:DG\ WHM SR]EDZLRQ\ MHVW XNáDG ]DVLODQLD ] DXWRPDW\F]Q SRODU\]DFM
EUDPNL]UHDOL]RZDQ\ZRSDUFLXRXNáDG]U\VZNWyU\P
U
GG
=
0
%UDPND MHVW QDGDO SRODU\]RZDQD VSDGNLHP QDSL FLD QD UH]\VWRU]H
R
S
:WHG\XNáDG]U\VDQD]\ZDP\XNáDGHP]DVLODQLD]DXWRPDW\F]Q\P
PLQXVHP
U
I R
GS
D
S
= −
D]U\VEXNáDGHP]DVLODQLD]DXWRPDW\F]Q\PSOXVHP
U
I R
GS
D
S
=
-HGQR (UyGáR ]DVLODQLD MHVW WDN*H Z\NRU]\VW\ZDQH Z XNáDG]LH
SRWHQFMRPHWU\F]Q\PSU]HGVWDZLRQ\PQDU\V
+
8
''
5
'
5
6
,
'
8
'6
>
,
,
6
'
=
5
8
*6
−
8
''
5
'
5
6
,
'
8
'6
<
,
,
6
'
=
5
*
8
*6
b)
a)
5
5\V 3RWHQFMRPHWU\F]Q\ XNáDG ]DVLODQLD WUDQ]\VWRUD 026)(7 ] NDQDáHP
]XER*DQ\PDW\SXQEW\SXS
8NáDG]U\VDRSLVXMUyZQDQLD
(
)
U
R
R
R
U
I R
U
U
I
R
R
GS
DD
D
S
DD
DS
D
D
S
=
+
−
=
+
+
2
1
2
QDWRPLDVWXNáDG]U\VE
(
)
U
R
R
R
U
I R
U
U
I
R
R
GS
DD
D
S
DD
DS
D
D
S
= −
+
+
= −
+
+
2
1
2
8NáDG\WHGDMZL NV]VZRERG GRERUXSXQNWXSUDF\
7UDQ]\VWRU\]áF]RZH-)(7NWyUHSUDFXM]D]XER*DQLHPPRJ
E\ü]DVLODQHSU]H]RPyZLRQHXNáDG\WUDQ]\VWRU-)(7]NDQDáHPW\SXQ
SU]H] XNáDG\ ] U\VD D D WUDQ]\VWRU ] NDQDáHP W\SX S SU]H]
XNáDG\]U\VEE
7UDQ]\VWRU\026]NDQDáHPZ]ERJDFDQ\PZREV]DU]HQDV\FHQLD
Z\PDJDMQDSL ü
U
GS
L
U
DS
RMHGQDNRZ\FK]QDNDFK]NDQDáHPW\SXQ
GRGDWQLFK]NDQDáHPSXMHPQ\FK3RGW\PZ]JO GHPVRQHSRGREQH
GRWUDQ]\VWRUyZELSRODUQ\FK7UDQ]\VWRU]NDQDáHPZ]ERJDFDQ\PW\SX
QPR*HE\ü]DVLODQ\SU]H]XNáDG]U\VDZNWyU\PZPLHMVFH(UyGáD
]DVLODQLD
−
U
GG
]DVWRVXMHP\ (UyGáR
+
U
GG
RUD] SU]H] XNáDG
]U\VD 5H]\VWDQFMH Z REX XNáDGDFK PXV] E\ü WDN GREUDQH DE\
QDSL FLH EUDPND (UyGáR E\áR GRGDWQLH $QDORJLF]QLH WUDQ]\VWRU
]NDQDáHP Z]ERJDFDQ\P W\SX S PR*H E\ü ]DVLODQ\ SU]H] XNáDG ]
U\VE Z NWyU\P Z PLHMVFH (UyGáD ]DVLODQLD
+
U
GG
]DVWRVXMHP\
(UyGáR
−
U
GG
RUD] SU]H] XNáDG ] U\VE 5H]\VWDQFMH Z REX
XNáDGDFK PXV] E\ü WDN GREUDQH DE\ QDSL FLH EUDPND (UyGáR E\áR
XMHPQH
=$6,/$1,(
,
67$%,/,=$&-$
381.7Ï:
35$&<
75$1=<6725Ï::8.à$'$&+6&$/21<&+
2JyOQDFKDUDNWHU\VW\NDREZRGyZ]DVLODQLDZXNáDGDFK
VFDORQ\FK
7HFKQRORJLD PRQROLW\F]QD VWZDU]D LGHDOQH ZDUXQNL GOD
Z\HNVSRQRZDQLD Z XNáDG]LH NRU]\VWQ\FK ZáDFLZRFL HOHPHQWyZ
Z\WZDU]DQ\FK QD Sá\WFH SyáSU]HZRGQLND Z W\P VDP\P SURFHVLH
WHFKQRORJLF]Q\P 1DOH*\ WX SU]HGH ZV]\VWNLP Z\PLHQLü GX*H
SRGRELHVWZR WUDQ]\VWRUyZ ]EOL*RQ WHPSHUDWXU ]áF] ZVSyáELH*QH
]PLDQ\ SDUDPHWUyZ WUDQ]\VWRUyZ RUD] UH]\VWDQFML SU]\ ]PLDQDFK
WHPSHUDWXU\RWRF]HQLD
:XNáDGDFKPRQROLW\F]Q\FKZ\VW SXMWH*LQQHRJUDQLF]HQLDQL*
Z XNáDGDFK G\VNUHWQ\FK : WHFKQRORJLL NáDGyZ ELSRODUQ\FK ]DNUHV
ZDUWRFL UH]\VWDQFML MHVW RJUDQLF]RQ\ GR NLONXG]LHVL FLX N
Ω
N
Ω
UH]\VWRU\ ED]RZH D SRQDGWR UR]U]XW\ ZDUWRFL UH]\VWDQFML V GX*H
±:WHFKQRORJLLXQLSRODUQHMSUDZLHZRJyOHQLHVWRVXMHVL
UH]\VWRUyZ SU]\ F]\P V RQH ]DVW SRZDQH (UyGáDPL SUGRZ\PL OXE
XNáDGDPL ÄOXVWHU SUGRZ\FK´ Z REZRGDFK SRODU\]DFML SUGHP VWDá\P
RUD] V VWRVRZDQH MDNR REFL*HQLD G\QDPLF]QH GOD VNáDGRZ\FK
]PLHQQ\FK
1LH PD PR*OLZRFL UHDOL]DFML NRQGHQVDWRUyZ R SRMHPQRFL
SU]HNUDF]DMFHM NLONDG]LHVLW S) GODWHJR XNáDG\ ZLHORVWRSQLRZH V
]UHJXá\ XNáDGDPL R VSU] *HQLDFK JDOZDQLF]Q\FK EH] NRQGHQVDWRUyZ
VHSDUXMF\FK L RGVSU] JDMF\FK : XNáDGDFK W\FK V]F]HJyOQHJR
]QDF]HQLDQDELHUDVWDáRüSXQNWyZSUDF\WUDQ]\VWRUyZJG\*LFK]PLDQ\
V Z]PDFQLDQH Z NROHMQ\FK VWRSQLDFK =H Z]JO GX QD EH]SRUHGQLH
VSU] *HQLDF] VWR]DFKRG]L SRWU]HED VWRVRZDQLD XNáDGyZ SU]HVXZDQLD
SR]LRPX QDSL FLD 6WDELOL]DFM SXQNWyZ SUDF\ ]DSHZQLD VL SU]H]
SRZV]HFKQH VWRVRZDQLH NRPSHQVDFML QLHOLQLRZHM NWyUD MHVW ]QDF]QLH
XáDWZLRQD G]L NL ]EOL*RQHM WHPSHUDWXU]H WUDQ]\VWRUyZ L GX*HPX
SRGRELHVWZXLFKFKDUDNWHU\VW\N
: PRQROLW\F]QHM WHFKQRORJLL ELSRODUQHM ] Z\MWNLHP W]Z
NRPSOHPHQWDUQHM WHFKQRORJLL ELSRODUQHM ] L]RODFM WOHQNRZ
FKDUDNWHU\VW\NLLSDUDPHWU\WUDQ]\VWRUyZSQSVJRUV]HQS
β
0
10
=
QL* WUDQ]\VWRUyZ QSQ QS
β
0
40
200
=
−
GODWHJR RJUDQLF]D VL LFK
VWRVRZDQLH
:XNáDGDFKPRQROLW\F]Q\FKSUDNW\F]QLHQLHPDPR*OLZRFLZ\NRQDQLD
HOHPHQWyZLQGXNF\MQ\FK
=H ZV]\VWNLPL HOHPHQWDPL XNáDGX VFDORQHJR ]ZL]DQH V ELHUQH
LF]\QQH HOHPHQW\ SDVR*\WQLF]H VNG NRQLHF]QRü VWRVRZDQLD
HOHPHQWyZ R ZDUWRFLDFK W\SRZ\FK D QLH PLQLPDOQ\FK
LPDNV\PDOQ\FK
3RGVWDZRZHEORNLZXNáDGDFK]DVLODQLDLVWDELOL]DFMLSXQNWyZ
SUDF\WUDQ]\VWRUyZELSRODUQ\FK
'UyGáDVWDáRSUGRZHOXVWUDSUGRZH
1D U\V SU]HGVWDZLRQR W\SRZ\ VSRVyE ]DVLODQLD WUDQ]\VWRUD
7
VWRVRZDQ\ZELSRODUQ\FKXNáDGDFKVFDORQ\FK
,
&
5
5
&
,
&
+
8
&&
,
%
,
%
,
5
8
%(
2
T
1
T
5\V 3U]\NáDG ]DVLODQLD WUDQ]\VWRUD
ZELSRODUQ\FKXNáDGDFKVFDORQ\FK
7UDQ]\VWRU 7
FKRFLD* SUDFXMH Z SRáF]HQLX GLRGRZ\P WR MHVW QDGDO
HOHPHQWHP DNW\ZQ\P SRQLHZD* ]QDMGXMH VL QD JUDQLF\ REV]DUX
DNW\ZQHJR
U
CB
=
0
7UDQ]\VWRU\7
L7
SUDFXMSU]\W\PVDP\PQDSL FLX
U
BE
PDMWDNLH
VDPHSUG\ED]
I
I
I
B
B
B
1
2
=
=
DW\PVDP\PUyZQLH*WDNLHVDPHSUG\
NROHNWRUyZ
I
I
I
C
C
C
1
2
=
=
'ODXNáDGX]U\VPR*QD]DSLVDü]DOH*QRFL
U
I R
U
I
I
I
I
I
CC
R
BE
R
C
B
C
B
=
+
=
+
=
+
1
2
2
5R]ZL]XMFXNáDGUyZQD]HZ]JO GXQD
,
&
RWU]\PXMHP\
I
U
U
R
I
U
U
R
I
C
CC
BE
B
CC
BE
C
=
−
−
=
−
−
2
2
0
β
-H*HOL
U
U
CC
BE
>>
β
0
2
>>
WRZ]yUXSUDV]F]DVL GRSRVWDFL
I
U
R
C
CC
≈
=DWHPZVSyáU] GQHSXQNWXSUDF\WUDQ]\VWRUD7
RNUHODM]DOH*QRFL
I
I
U
R
U
U
I R
U
R
R
C
C
CC
CE
CC
C
C
CC
C
2
2
2
1
=
≈
=
−
≈
−
3RQLHZD* WHPSHUDWXURZH ]PLDQ\ UH]\VWDQFML
R
C
L
R
Z XNáDG]LH
VFDORQ\P PRJ E\ü ZVSyáELH*QH WHPSHUDWXURZ\ ZVSyáF]\QQLN
VWRVXQNX
R
R
C
MHVWEDUG]RPDá\]DWHPSXQNWSUDF\WUDQ]\VWRUD7
QLH
]DOH*\ RG ]PLDQ WHPSHUDWXU\ D MHG\QLH RG Uy*QLF\ SDUDPHWUyZ
WUDQ]\VWRUyZ7
L7
7HQ SURVW\ VSRVyE Z\PXV]DQLD SUGX MHGQHJR WUDQ]\VWRUD SU]H]
GUXJL VWDQRZL SRGVWDZ UHDOL]DFML (UyGHá VWDáRSUGRZ\FK Z XNáDGDFK
VFDORQ\FK
6FKHPDW LGHRZ\ QDMSURVWV]HJR D SU]\ W\P SRGVWDZRZHJR (UyGáD
VWDáRSUGRZHJR]WUDQ]\VWRUDPLELSRODUQ\PLSU]HGVWDZLRQRQDU\V
,
,
2
&
=
,
&
,
%
,
%
,
5()
8
%(
2
T
1
T
8
&(
5\V 3RGVWDZRZH (UyGáR
VWDáRSUGRZHOXVWURSUGRZH
=DNáDGDMF MDN SRSU]HGQLR LGHQW\F]QRü WUDQ]\VWRUyZ RUD] SRPLMDMF
ZSá\ZQDSL FLD
U
CE 2
WUDQ]\VWRUD7
QDMHJRSUGNROHNWRUDZREV]DU]H
DNW\ZQ\PPR*HP\QDSLVDü
I
I
I
I
I
I
I
I
C
B
B
C
B
O
C
REF
1
1
2
1
1
0
2
2
+
+
=
+
=
+
=
β
VWGRWU]\PXMHP\
I
I
I
O
REF
C
=
+
=
1 2
0
1
β
D
-H*HOL
β
0
1
>>
WR
I
I
O
REF
≈
E
3UG Z\MFLRZ\ (UyGáD
I
O
MHVW UyZQ\ Z SU]\EOL*HQLX SUGRZL
RGQLHVLHQLD
I
REF
Z\PXV]RQHPX SU]H] (UyGáR RGQLHVLHQLD DQJ
UHIHUHQFH FXUUHQW 7HQ URG]DM (UyGáD VWDáRSUGRZHJR QRVL QD]Z
]ZLHUFLDGáDSUGRZHJROXEOXVWUDSUGRZHJRDQJFXUUHQWPLUURU
:Sá\Z QDSL FLD
U
CE 2
QD SUG (UyGáD RNUHOLP\ QD SRGVWDZLH
FKDUDNWHU\VW\NLZ\MFLRZHMWUDQ]\VWRUD7
U\V
0
X
&(2
8
&(2
(
)
, 8
&
&(
2
2
(
)
, 8
&
&(
2
2
0
=
L
&2
8
$
QDSL FLH (DUO\HJR
U]HF]\ZLVWD UH]\VWDQFMD
(UyGáD SUGX
QLHVNRF]RQD UH]\VWDQFMD
(UyGáD SUGX
$
%
5\V:Sá\ZQDSL FLD
8
&(
QDSUG(UyGáD
3U]\ ]PLDQLH QDSL FLD
U
CE 2
]PLHQLD VL SUG NROHNWRUD
I
C 2
ZVNXWHN
]MDZLVNDPRGXODFMLV]HURNRFLED]\
I
I
U
U
U
C
S
BE
T
CE
A
2
2
1
=
−
exp
ϕ
.RU]\VWDMF]SURSRUFMLPR*HP\QDSLVDü
(
)
(
)
I
U
U
U
I
U
U
C
CE
A
CE
C
CE
A
2
2
2
2
2
0
+
=
=
&]\OL
(
)
(
)
I
I
U
I
U
U
U
O
C
CE
C
CE
CE
A
=
=
=
+
2
2
2
2
2
0 1
=HZ]JO GXQDUy*QHQDSL FLD
U
CE
WUDQ]\VWRUyZUyZQLH*SUG\
LFK NROHNWRUyZ QLH V MHGQDNRZH
I
I
C
C
1
2
≠
=DNáDGDMF UyZQRü
SUGyZSU]\]HURZ\PQDSL FLX
U
CE
VWRVXQHNW\FKSUGyZZ\QRVL
I
I
U
U
U
U
U
U
C
C
CE
A
CE
A
CE
A
2
1
2
1
2
1
1
1
=
+
+
≈ +
5y*QLF]NRZDUH]\VWDQFMDZ\MFLRZD(UyGáDZ\QRVL
(
)
(
)
r
u
i
U
U
I
U
U
I
U
r
o
CE
C
A
CE
C
CE
A
C
CE
ec
=
=
+
≈
≈
∆
∆
2
2
2
2
2
2
2
: OXVWU]H SUGRZ\P LVWQLHMH PR*OLZRü Z\PXV]HQLD Z GUXJLP
WUDQ]\VWRU]HSUGXUy*QHJRRGSUGXRGQLHVLHQLD=DOHWDWDMHVWEDUG]R
ZD*QD ]H Z]JO GyZ SUDNW\F]Q\FK -H*HOL SRZLHU]FKQLD ]áF] HPLWHU
ED]D WUDQ]\VWRUyZ 7
L 7
Z XNáDG]LH ] U\V V Uy*QH L Z\QRV]
RGSRZLHGQLR
S
1
L
S
2
WRVWRVXQHNSUGyZNROHNWRUyZW\FKWUDQ]\VWRUyZ
MHVWUyZQ\VWRVXQNRZLSRZLHU]FKQLLFK]áF]HPLWHURZ\FK
=DWHP
I
I
S
S
C
C
1
2
1
2
=
3RQLHZD*
I
I
I
I
I
I
I
C
REF
B
B
REF
C
C
1
1
2
1
0
2
0
=
−
−
=
−
−
β
β
WRSRSRGVWDZLHQLXGRRWU]\PXMHP\
I
I
I
S S
S S
O
C
REF
=
=
+ +
2
2
1
2
1
0
1
1
β
-H*HOL
S
S
1
2
=
WR Z]yU SU]\MPXMH SRVWDü QDWRPLDVW JG\
β
0
2
1
1
>> +
S S
WR
I
I
S
S
O
REF
≈
2
1
:\UD*HQLH
S S
S S
S
S
2
1
2
1
0
2
1
1
1
+ +
≈
β
QD]\ZD VL Z]PRFQLHQLHP OXVWUD
SUGRZHJR
1DU\VSU]HGVWDZLRQR]HVSyáOXVWHUSUGRZ\FKZNWyU\PWHQ
VDP SUG RGQLHVLHQLD Z\PXV]D MHGQRF]HQLH ]JRGQLH ] ]DOH*QRFL
XVWDORQHSUG\Z\MFLRZHZZLHOXWUDQ]\VWRUDFK
,
&
1,
%
,
%
,
5()
T
1
T
2
T
3
T
T
N
,
1
,
2
,
3
,
1
5\V=HVSyáOXVWHU
SUGRZ\FK
=Dáy*P\ GOD XSURV]F]HQLD UR]ZD*D *H ZV]\VWNLH WUDQ]\VWRU\ V
MHGQDNRZHPDMW VDPSRZLHU]FKQL ]áF]DHPLWHURZHJR
:WHG\]JRGQLH]HVFKHPDWHPRWU]\PXMHP\
(
)
I
I
N
I
REF
C
B
=
+
+
1
6WG
I
I
I
I
I
N
N
O
REF
1
2
0
1
1
= =
=
=
=
+
+
β
'OD GX*\FK ZDUWRFL
β
0
L QLH]E\W GX*HM OLF]E\
N
ZV]\VWNLH SUG\
Z\MFLRZH V Z SU]\EOL*HQLX UyZQH
I
REF
=Uy*QLFRZDQLH W\FK SUGyZ
RWU]\PXMHP\SU]\Uy*Q\FKSRZLHU]FKQLDFK]áF]HPLWHURZ\FK
: SU]\SDGNX ]DVWRVRZDQLD WUDQ]\VWRUyZ SQS R PDáHM ZDUWRFL
β
0
β
0
10
≈
DOER SU]\ SRáF]HQLX UD]HP ]E\W GX*HM LORFL ED]
WUDQ]\VWRUyZUHDOL]XMF\FK]HVSyáOXVWHUSUGRZ\FKSRMDZLDMVL FRUD]
ZL NV]H Uy*QLFH SRPL G]\ SUGHP RGQLHVLHQLD
I
REF
D SUGHP
Z\MFLRZ\P
1D U\V SU]HGVWDZLRQR ]PRG\ILNRZDQ\ XNáDG (UyGáD
VWDáRSUGRZHJR OXVWUD SUGRZHJR R ]PQLHMV]RQ\P ZSá\ZLH SUGyZ
ED]
,
,
2
&
=
,
&
,
%
,
%
,
5()
8
%(
2
T
1
T
,
( 3
,
%3
T
3
5\V 'UyGáR VWDáRSUGRZH
OXVWUR SUGRZH R ]PQLHMV]RQ\P
ZSá\ZLHSUGyZED]
3UG\HPLWHUDLED]\WUDQ]\VWRUD7
Z\QRV]
I
I
I
I
I
I
E
B
B
C
C
O
3
1
2
1
2
0
0
2
=
+
=
+
=
β
β
(
)
I
I
I
B
E
O
3
3
0
0
0
1
2
1
=
+
=
+
β
β β
3R]VXPRZDQLXSUGyZZZ (OHNROHNWRUDWUDQ]\VWRUD7
RWU]\PXMHP\
(
)
I
I
I
REF
C
O
−
−
+
=
1
0
0
2
1
0
β β
6WG
(
)
I
I
O
REF
=
+
+
1
2
1
0
0
β β
1DZHWGODPDá\FKZDUWRFL
β
0
ZDUXQHN
(
)
β β
0
0
1
2
+ >>
MHVWVSHáQLRQ\
DW\PVDP\PSUGZ\MFLRZ\
I
O
QLHZLHOHUy*QLVL RGSUGXRGQLHVLHQLD
I
REF
3RGREQHZáDFLZRFLGRRSLVDQ\FKZ\*HMSRVLDGD(UyGáRSUGRZH
OXVWURSUGRZH:LOVRQDU\VZNWyU\PUHODFM SRPL G]\SUGHP
Z\MFLRZ\P
I
O
DSUGHPRGQLHVLHQLD
I
REF
RSLVXMHLGHQW\F]QD]DOH*QRü
MDN
,
2
,
5()
8
%(
2
T
T
3
1
T
U
R
8
%(
5\V /XVWUR SUGRZH
:LOVRQD
=H Z]JO GX QD NDVNRGRZH SRáF]HQLH WUDQ]\VWRUyZ 7
L 7
OXVWUR
SUGRZH :LOVRQD SRVLDGD ]QDF]QLH ZL NV] G\QDPLF]Q UH]\VWDQFM
Z\MFLRZQL*XNáDG\]SRMHG\QF]\PLWUDQ]\VWRUDPL
5H]\VWDQFMDWDZ\QRVL
r
U
I
o
A
O
≈
β
0
2
3RQDGWR OXVWUR SUGRZH :LOVRQD FKDUDNWHU\]XMH VL OHSV]\PL
ZáDFLZRFLDPL
F] VWRWOLZRFLRZ\PL
V]HURNRSDVPRZ\PL
ZSRUyZQDQLXGRLQQ\FKOXVWHUSUGRZ\FK
2JyOQPHWRG]DSHZQLDMF]ZL NV]HQLHUH]\VWDQFMLZ\MFLRZHM
(UyGáD VWDáRSUGRZHJR OXVWUD SUGRZHJR MHVW EXGRZDQLH XNáDGyZ
NDVNRGRZ\FK
3RGVWDZRZ\ XNáDG NDVNRGRZHJR OXVWUD SUGRZHJR SU]HGVWDZLRQR QD
U\V
7UDQ]\VWRU 7
SUDFXMH SRGREQLH MDN WUDQ]\VWRU 7
SU]\ QDSL FLX
U
CB
=
0
'\QDPLF]QD UH]\VWDQFMD Z\MFLRZD WHJR WUDQ]\VWRUD UyZQD
r
U
I
U
I
ce
A
C
A
O
4
4
=
≈
ZQRVL XMHPQH VSU] *HQLH ]ZURWQH SUGRZH
V]HUHJRZH Z REZRG]LH HPLWHUD WUDQ]\VWRUD 7
SRZRGXMF ]QDF]QH
]ZL NV]HQLHZ\SDGNRZHMG\QDPLF]QHMUH]\VWDQFMLZ\MFLRZHMXNáDGXGR
ZDUWRFL
r
U
I
o
A
O
≈
β
0
,
2
,
5()
8
%(
4
T
T
2
3
T
U
R
8
%(
1
T
5\V .DVNRGRZH (UyGáR
VWDáRSUGRZHOXVWURSUGRZH
']L NL WDN GX*HM ZDUWRFL G\QDPLF]QHM UH]\VWDQFML Z\MFLRZHM
SUGZ\MFLRZ\(UyGáDQLHXOHJD]PLDQLHSRGZSá\ZHP]PLHQLDMFHJR
VL QDSL FLDQDZ\MFLX
6WDáRSUGRZH(UyGáDRGQLHVLHQLD
2PyZLRQH Z SRSU]HGQLP UR]G]LDOH (UyGáD VWDáRSUGRZH
QD]\ZDQH UyZQRF]HQLH OXVWUDPL SUGRZ\PL UR]ZD*DQR JáyZQLH SRG
NWHP GRNáDGQRFL ] MDN Z\PXV]DQ\ MHVW SUG Z\MFLRZ\
I
O
SU]H]
SUG RGQLHVLHQLD
I
REF
1LH ]DMPRZDQR VL QDWRPLDVW Z MDNL VSRVyE
Z\WZRU]\ü SUG RGQLHVLHQLD R PDáHM ZUD*OLZRFL QD ]PLDQ\ QDSL FLD
]DVLODMFHJRLWHPSHUDWXU\
: SU]\SDGNX UHDOL]DFML VWDáRSUGRZ\FK (UyGHá RGQLHVLHQLD
SRGVWDZRZ VSUDZ MHVW VWDELOL]DFMD QDSL FLRZD WDN DE\ Z\MFLRZH
SUG\ RGQLHVLHQLD E\á\ QLH]DOH*QH RG QDSL ü ]DVLODMF\FK 1D U\V
SU]HGVWDZLRQR SURVWH SU]\NáDG\ WDNLFK (UyGHá Z NWyU\FK Z\VW SXMH
MHGQDNVáDED]DOH*QRüSUGXZ\MFLRZHJRRGQDSL FLD]DVLODMFHJR
,
2
,
8
%(
8
%(
3
T
2
T
5
5
8
%(
T
1
a)
,
2
,
8
%(
2
T
5
5
T
1
b)
8
%(
+
8
&&
+
8
&&
5\V'UyGáD VWDáRSUGRZH
QLH]DOH*QHRGQDSL FLD]DVLODQLD
=DQLHGEXMFSUGED]\WUDQ]\VWRUD7
GODREXXNáDGyZPR*QDQDSLVDü
I
U
R
R
I
I
O
BE
T
S
≈
=
2
2
1
ϕ
ln
3RQLHZD*SUG
I
1
]DOH*\RGQDSL FLD
U
CC
I
U
U
R
CC
BE
1
1
2
=
−
]DWHP SUG (UyGáD SRUHGQLR MHVW ORJDU\WPLF]Q IXQNFM QDSL FLD
]DVLODQLD
1D SRGVWDZLH ]DOH*QRFL PR*HP\ Z\]QDF]\ü
ZUD*OLZRü
I
O
SUGXQD]PLDQ\QDSL FLD
U
CC
S
U
I
I
U
I R
U
I
CC
O
O
CC
T
O
CC
O
=
=
∂
∂
ϕ
2
:DG XNáDGyZ ] U\V MHVW VLOQD ]DOH*QRü QDSL FLD
U
BE
RG
WHPSHUDWXU\
'OD RFHQ\ ZUD*OLZRFL (UyGáD QD ]PLDQ\ WHPSHUDWXU\ GHILQLXMH VL
WHPSHUDWXURZ\ZVSyáF]\QQLNZ]JO GQHM]PLDQ\SUGX(UyGáD
( )
TW I
I
I
T
T
T
I
I
T
T
S
O
O
O
O
O
T
I
O
=
=
=
1
1
1
∂
∂
∂
∂
:VSyáF]\QQLN
TW
Z\UD*DVL ZMHGQRVWNDFK
°
&OXEZSSP
°
&
$QDOL]RZDQ\ XNáDG FKDUDNWHU\]XMH VL GRü GX* ZUD*OLZRFL
I
O
QD
]PLDQ\
U
CC
Z\QRV]F RN RUD] GX*\P ZVSyáF]\QQLNLHP
( )
TW I
O
NWyU\PR*QDRV]DFRZDüRNSSP
°
&
/HSV]\P UR]ZL]DQLHP MHVW (UyGáR VWDáRSUGRZH SU]HGVWDZLRQH QD
U\VQD]\ZDQH(UyGáHP:LGODUD
,
2
,
&
,
%
,
%
5
8
%(
2
T
1
T
8
%(
,
+
8
&&
5
5\V 6WDáRSUGRZH (UyGáR
:LGODUD
6WRVXMFSUDZR.LUFKKRIIDRWU]\PXMHP\
U
U
I R
I
I
I
I
I R
BE
BE
O
T
C
ES
T
O
ES
O
1
2
2
1
1
2
2
0
−
−
=
−
−
=
ϕ
ϕ
ln
ln
-H*HOLWUDQ]\VWRU\VMHGQDNRZHWR
I
I
ES
ES
1
2
=
LZWHG\
I
R
I
I
O
T
C
O
= ϕ
2
1
ln
=DNáDGDMF I
I
I
B
B
C
1
2
1
,
<<
RUD]
U
U
BE
CC
1
<<
PR*HP\ Z\]QDF]\ü
SU]\EOL*RQZDUWRüSUGXNROHNWRUD
I
C1
I
I
U
U
R
U
R
C
CC
BE
CC
1
1
1
1
1
≈ =
−
≈
3RGVWDZLDMFGRRWU]\PXMHP\
I
R
U
I R
O
T
CC
O
≈ ϕ
2
1
ln
-DNZ\QLND]UyZQDQLDVSDGHNQDSL FLD
I R
O
2
MHVWUyZQ\Uy*QLF\
U
U
BE
BE
1
2
−
FR R]QDF]D *H QDZHW SU]\ VWRVRZDQLX QLH]E\W GX*\FK
ZDUWRFL
R
2
SUG
I
O
PR*HE\üPDá\-HVWWRZáDFLZRü(UyGáD:LGODUD
Z\NRU]\VW\ZDQD]DZV]HJG\Z\PDJDQ\MHVWPDá\SUG(UyGáD
3UG (UyGáD
I
O
MHVW ORJDU\WPLF]Q IXQNFM QDSL FLD ]DVLODQLD U
CC
UyZ D MHJR ZUD*OLZRü QD ]PLDQ\ WHPSHUDWXURZH MHVW PQLHMV]D
QL*ZXNáDG]LH]U\V
1D U\V SU]HGVWDZLRQR VWDáRSUGRZH (UyGáR RGQLHVLHQLD ]
VDPRF]\QQSRODU\]DFMQDSL FLHP
8
%(
ZNWyU\PSUG\Z\MFLRZHV
FDáNRZLFLHQLH]DOH*QHRGQDSL FLD]DVLODMFHJRDQJVHOIELDVLQJ
8
%(
UHIHUHQFH>@
,
2
,
&
5
,
+
8
&&
5
3
T
,
2
5
2
T
4
T
5
T
6
T
1
T
'
'
'
'
'
a)
,
,
&
3
,
5
,
,
&
7
2
=
ϕ
OQ
,
,
&
=
b)
5\V6WDáRSUGRZH(UyGáRRGQLHVLHQLD]VDPRF]\QQSRODU\]DFMQDSL FLHP
8
%(
DVFKHPDWEORNRZ\EFKDUDNWHU\VW\NDSU]HMFLRZD
8NáDG PD GZD Z\MFLD MHGQR W\SX HPLV\MQHJR Z NWyU\P SUG (UyGáD
Z\Sá\ZD RG GRGDWQLHJR ELHJXQD QDSL FLD ]DVLODQLD RUD] GUXJLH W\SX
DEVRUEF\MQHJR Z NWyU\P SUG (UyGáD ZSá\ZD GR PDV\ XNáDGX DQJ
FXUUHQWVRXUFHVLQN
7UDQ]\VWRU\7
7
WZRU]XNáDGMDNQDU\VE]DWHPSUGNROHNWRUD
I
C1
GDQ\MHVWZ]RUHP/XVWURSUGRZH]áR*RQH]WUDQ]\VWRUyZ7
7
Z\PXV]DUyZQRüSUGyZ
I
I
C1
1
=
5R]ZL]DQLHPREXUyZQDMDN
WR SU]HGVWDZLRQR JUDILF]QLH QD U\VE MHVW SXQNW 3 OXE SXQNW
]QDMGXMF\ VL Z SRF]WNX XNáDGX ZVSyáU] GQ\FK F]\OL
I
I
C1
1
0
= =
$E\ Z\HOLPLQRZDü PR*OLZRü XVWDOHQLD VL QLHSR*GDQHJR SXQNWX
SUDF\ SU]\
I
C1
0
=
XNáDG SRVLDGD VSHFMDOQH ]DEH]SLHF]HQLH ] GLRGDPL
'
'
WUDQ]\VWRU\ Z SRáF]HQLX GLRGRZ\P NWyUHJR ]DGDQLHP MHVW
VSRZRGRZDQLH SU]HSá\ZX SUGX SU]H] UH]\VWRU 5
ZyZF]DV JG\
I
C1
0
=
= FKZLO JG\ RVLJQL W\ ]RVWDQLH ZáDFLZ\ SXQNW SUDF\ 3
GLRGD'
VSRODU\]RZDQDMHVW]DSRURZRLXNáDG]DEH]SLHF]DMF\]RVWDMH
RGáF]RQ\RGUHV]W\XNáDGX
'UyGáDQDSL FLDRGQLHVLHQLD
'UyGáD QDSL FLD RGQLHVLHQLD DQJ YROWDJH UHIHUHQFHV SRZLQQ\
SRVLDGDüPDáUH]\VWDQFM Z\MFLRZRUD]SRZLQQ\E\üQLHZUD*OLZHQD
]PLDQ\ QDSL FLD ]DVLODQLD L WHPSHUDWXU\ : SU]\SDGNX UHDOL]DFML W\FK
(UyGHá JáyZQ\P ]DGDQLHP MHVW VWDELOL]DFMD WHPSHUDWXURZD .RV]WHP
UH]\JQDFML ] EDUG]R PDáHM UH]\VWDQFML Z\MFLRZHM Z\PDJD VL DE\
WHPSHUDWXURZ\ ZVSyáF]\QQLN QDSL FLD RGQLHVLHQLD
(
)
TW U
REF
E\á
PQLHMV]\QL*SSP
°
&
3URVWHSU]\NáDG\(UyGHáQDSL FLRZ\FKSU]HGVWDZLRQHQDU\V
LOXVWUXMMHG\QLHPHWRG\XVWDODQLDQDSL FLDZ\MFLRZHJRR]PQLHMV]RQHM
ZUD*OLZRFL QD ]PLDQ\ QDSL FLD ]DVLODQLD EH] VWDELOL]DFML
WHPSHUDWXURZHM
1DSL FLH QD Z\MFLX RUD] G\QDPLF]QD UH]\VWDQFMD Z\MFLRZD XNáDGX ]
U\VDZ\QRV]
U
U
U
REF
Z
BE
=
+
r
U
I
U
I
r
I
o
Z
BE
z
T
=
+
= +
∂
∂
∂
∂
ϕ
1DSL FLH
U
BE
QD WUDQ]\VWRU]H 7
Z SRáF]HQLX GLRGRZ\P R XMHPQ\P
ZVSyáF]\QQLNXWHPSHUDWXURZ\PF] FLRZRNRPSHQVXMHWHPSHUDWXURZH
]PLDQ\QDSL FLD
U
Z
RGRGDWQLPZVSyáF]\QQLNXWHPSHUDWXURZ\P
+
8
&&
1
T
'=
a)
8
5()
1
T
2
T
N
T
8
5()
+
8
&&
+
8
&&
8
5()
1
T
,
,
,
%
5
5
8
%(
,
,
,
b)
c)
5\V 3URVWH SU]\NáDG\ (UyGHá QDSL FLRZ\FK D ] GLRG =HQHUD E R QDSL FLX
Z\MFLRZ\P
18
%(
FPQR*QLN
8
%(
'UyGáRQDSL FLRZHQDU\VESRVLDGDQDSL FLHZ\MFLRZH
U
NU
REF
BE
=
RUD]G\QDPLF]QUH]\VWDQFM Z\MFLRZ
r
N
U
I
N
I
o
BE
T
=
=
∂
∂
ϕ
:DG WHJR XNáDGX MHVW EDUG]R GX*D ZDUWRü WHPSHUDWXURZHJR
ZVSyáF]\QQLNDQDSL FLDZ\MFLRZHJR
∂
∂
∂
∂
U
T
N
U
T
N mV
C
REF
BE
=
≈ −
2
$
5\VFSU]HGVWDZLDDOWHUQDW\ZQHUR]ZL]DQLHQD]\ZDQHPQR*QLNLHP
U
BE
Z NWyU\P SU]\ SRPLQL FLX SUGX ED]\ Z VWRVXQNX GR SUGX
G]LHOQLND
I
I
1
2
≈
RWU]\PXMHP\
U
R
R
R
U
BE
REF
=
+
2
1
2
6WG
U
R
R
U
REF
BE
= +
1
1
2
'\QDPLF]QDUH]\VWDQFMDZ\MFLRZD(UyGáDZ\QRVL
r
R
R
R
g R
o
m
=
+
+
1
0
1
2
2
β
1DU\VSU]HGVWDZLRQR(UyGáRQDSL FLDRGQLHVLHQLD]WHPSHUDWXURZ
NRPSHQVDFMQDSL ü
U
Z
L
U
BE
>@
'
=
1
T
3
T
8
5()
+
8
&&
5
8
%(
,
2
T
8
%(
8
=
8
%(
5
5\V 'UyGáR QDSL FLD
RGQLHVLHQLD ] WHPSHUDWXURZ
NRPSHQVDFMQDSL ü
8
=
L
8
%(
1DSL FLHZ\MFLRZH(UyGáDZ\QRVL
(
)
(
)
U
U
U
U
R
R
R
R U
R
R U
R
R
REF
BE
Z
BE
Z
BE
=
+
−
+
=
+
−
+
3
2
2
1
2
2
1
2
1
2
=DQLHGEXMF ZSá\Z WHPSHUDWXURZHJR ZVSyáF]\QQLND VWRVXQNX
UH]\VWRUyZ
R R
1
2
QD QDSL FLH Z\MFLRZH ZDUXQHN NRPSHQVDFML
RWU]\PXMHVL SU]\UyZQXMFGR]HUDSRFKRGQ
dU
dT
REF
F]\OL
dU
dT
R
R
R
dU
dT
R
R
R
R
dU
dT
REF
Z
BE
=
+
+
−
+
=
2
1
2
1
2
1
2
2
0
=UyZQDQLDRWU]\PXMHP\ZDUXQHNNRPSHQVDFML
R
R
dU
dT
dU
dT
Z
BE
1
2
2
= +
=DNáDGDMFW\SRZHZDUWRFL
dU
dT
mV
C
Z
= +
3
$
RUD]
dU
dT
mV
C
BE
= −
2
$
RWU]\PXMHP\
R
R
1
5
0 5
/
,
=
U
U
REF
Z
=
/ 3
:WHFKQLFHXNáDGyZPRQROLW\F]Q\FKEDUG]RF] VWRZ\NRU]\VWXMH
VL RJyOQ ]DVDG UHDOL]DFML (UyGáD QDSL FLD RGQLHVLHQLD SROHJDMF QD
NRPSHQVDFML XMHPQHJR WHPSHUDWXURZHJR ZVSyáF]\QQLND QDSL FLD
U
BE
SU]H] GRGDWQL ZVSyáF]\QQLN QDSL FLD
ϕ
T
=DVDG W ]LOXVWURZDQR
VFKHPDWHPEORNRZ\PQDU\V>@
T
8
8
.
5()
%(
7
=
+
ϕ
8
%(
8
%(
,
.
7
ϕ
.
ϕ
7
ϕ
7
*HQHUDWRU
+
5\V2JyOQD]DVDGDUHDOL]DFML(UyGáDQDSL FLDRGQLHVLHQLDW\SXEDQGJDS
1DSL FLHRGQLHVLHQLD
8
5()
MHVWUyZQH
8
8
.
5()
%(
7
=
+
ϕ
:DUXQHNNRPSHQVDFMLRWU]\PXMHVL JG\
G8
G7
G8
G7
.
G
G7
5()
%(
7
=
+
=
ϕ
= UyZQDQLD PR*HP\ Z\]QDF]\ü Z\PDJDQ ZLHONRü
QLH]DOH*QHJRRGWHPSHUDWXU\ZVSyáF]\QQLND.
.
8
7
7
%(
7
=
∆
∆
∆
∆
ϕ
8Z]JO GQLDMFW\SRZHZDUWRFL
G8
G7
P9 &
%(
= −
$
RUD]
∆
∆
ϕ
7
7
P9 &
= +
$
RWU]\PXMHP\
.
=
RUD]
8
5()
=
9
3RQLHZD* ZDUWRü QDSL FLD RGQLHVLHQLD MHVW ]EOL*RQD GR ZDUWRFL
QDSL FLDEDULHU\SRWHQFMDáXNU]HPXGODWHJRXNáDG\UHDOL]RZDQHZJWHM
]DVDG\ V QD]\ZDQH XNáDGDPL ] EDULHU SRWHQFMDáX DQJ EDQG JDS
UHIHUHQFHFLUFXLWV
1D U\V SU]HGVWDZLRQR SU]\NáDGRZH UR]ZL ]DQLH (UyGáD
QDSL FLDRGQLHVLHQLDW\SXEDQGJDS
7UDQ]\VWRU\ 7
L 7
WZRU] VWDáRSUGRZH (UyGáR :LGODUD 6SDGHN
QDSL FLDQDUH]\VWDQFML
R
3
SU]\SRPLQL FLXSUGyZED]Z\QRVL
I R
U
U
I
I
C
BE
BE
T
C
C
2
3
1
2
1
2
=
−
=
ϕ
ln
,
&
8
2
T
5
5
8
%(
+
8
&&
5
8
%(
1
T
,
&
3
T
8
5()
,
5\V
3U]\NáDGRZD
UHDOL]DFMD (UyGáD RGQLHVLHQLD
W\SXEDQGJDS
:\]QDF]DMF ] UyZQDQLD SUG
I
C 2
SU]\
β
0
1
>>
PR*HP\
Z\]QDF]\üQDSL FLHQDUH]\VWDQFML
R
2
U
I R
R
R
I
I
C
T
C
C
2
2
2
2
3
1
2
=
=
ϕ
ln
1DSL FLHRGQLHVLHQLDMHVWUyZQH
U
U
U
U
R
R
I
I
REF
BE
BE
T
C
C
=
+
=
+
2
3
3
2
3
1
2
ϕ
ln
3RUyZQXMFLRWU]\PXMHP\
K
R
R
I
I
C
C
=
2
3
1
2
ln
6WRVXQHNSUGyZ
I
I
C
C
1
2
MHVWQLH]DOH*Q\RGWHPSHUDWXU\
$E\ X]\VNDü SHáQ NRPSHQVDFM WHPSHUDWXURZ QDSL FLD
U
REF
WR
GRELHUDMF QS GOD 7 .
I
I
C
C
1
2
10
=
PXVLP\ ]DSHZQLü
R R
2
3
10 2
=
,
3RGVWDZRZHEORNLZXNáDGDFK]DVLODQLDLVWDELOL]DFMLSXQNWyZ
SUDF\WUDQ]\VWRUyZXQLSRODUQ\FK
'UyGáDVWDáRSUGRZHOXVWUDSUGRZH
8NáDG\(UyGHáVWDáRSUGRZ\FKOXVWHUSUGRZ\FKUHDOL]RZDQHV
Z WHFKQRORJLL 026 QD EDUG]R SRGREQ\FK ]DVDGDFK MDN Z WHFKQRORJLL
ELSRODUQHM
1D U\V SU]HGVWDZLRQR SURVWH (UyGáR VWDáRSUGRZH OXVWUR
SUGRZH ] WUDQ]\VWRUDPL 026)(7 ] NDQDáHP Z]ERJDFDQ\P W\SX Q
VWDQRZLFHRGSRZLHGQLNXNáDGX]U\V
,
,
2
'
=
,
'
,
5()
8
*6
8
'6
2
M
1
M
5\V 3URVWH OXVWUR
SUGRZH
-H*HOLWUDQ]\VWRU\0
0
SUDFXMZREV]DU]HQDV\FHQLDWR
(
) (
)
(
) (
)
I
K W L U
U
U
I
K W L U
U
U
D
N
GS
T
DS
D
N
GS
T
DS
1
1
1
2
1
2
2
2
2
2
1
1
=
−
+
=
−
+
'
'
λ
λ
3RQLHZD*REDWUDQ]\VWRU\SUDFXMSU]\W\PVDP\PQDSL FLX SRGáR*H
(UyGáR
U
BS
GODWHJRPDMWHVDPHQDSL FLDSURJRZH
U
U
U
T
T
T
1
2
=
=
=DWHPQDSRGVWDZLHUyZQDRWU]\PXMHP\
(
) (
)
(
) (
)
I
I
I
I
K W L
U
K W L
U
O
REF
D
D
N
DS
N
DS
=
=
+
+
2
1
2
2
1
1
1
1
'
'
λ
λ
3U]\ SRPLQL FLX HIHNWX PRGXODFML GáXJRFL NDQDáX VWRVXQHN SUGyZ
I
I
O
REF
]DOH*\W\ONRRGUR]PLDUyZWUDQ]\VWRUyZ
(
)
(
)
I
I
W L
W L
O
REF
=
2
1
'\QDPLF]QDUH]\VWDQFMDZ\MFLRZD(UyGáDZ\QRVL
r
U
I
I
I
r
o
DS
D
D
O
ds
=
=
=
=
∆
∆
2
2
2
2
1
1
λ
λ
$E\ WUDQ]\VWRU 0
SUDFRZDá ]DZV]H Z REV]DU]H QDV\FHQLD PXVL E\ü
VSHáQLRQ\ZDUXQHN
U
U
U
U
GS
T
DS
−
>
=
2
0 min
,
1
,
5()
2
M
M
,
2
3
M
,
1
N
M
5\V=HVSyá OXVWHU
SUGRZ\FK
1DU\VSU]HGVWDZLRQR]HVSyáOXVWHUSUGRZ\FKZNWyU\PWHQ
VDP SUG ZHMFLRZ\ OXVWUD
I
REF
Z\PXV]D MHGQRF]HQLH 1 SUGyZ
Z\MFLRZ\FK]JRGQLH]UHODFM
1D U\V SU]HGVWDZLRQR VFKHPDW\ OXVWHU SUGRZ\FK :LOVRQD
NWyUH ] XZDJL QD NDVNRGRZH SRáF]HQLH GZyFK WUDQ]\VWRUyZ Z
REZRG]LH Z\MFLRZ\P FKDUDNWHU\]XM VL GX*R ZL NV] UH]\VWDQFM
Z\MFLRZQL*UH]\VWDQFMDSURVWHJROXVWUD
,
5()
2
M
1
M
3
M
8
*6
,
2
,
5()
2
M
1
M
3
M
,
2
4
M
a)
b)
5\V/XVWUD:LOVRQDD]PRG\ILNRZDQHE
: XNáDG]LH ]PRG\ILNRZDQ\P QD U\VE G]L NL GRGDQLX
WUDQ]\VWRUD0
ZREXJDá ]LDFKWUDQ]\VWRU\SUDFXMSU]\WDNLFKVDP\FK
QDSL FLDFK
U
DS
FR PD LVWRWQH ]QDF]HQLH GOD VSHáQLHQLD UHODFML
I
I
REF
O
=
=SRUyGZLHOXUy*Q\FKUR]ZL]DXNáDGRZ\FKOXVWHUSUGRZ\FKOXVWUD
:LOVRQD
Z\Uy*QLDM
VL
NRU]\VWQLHMV]\PL
ZáDFLZRFLDPL
F] VWRWOLZRFLRZ\PLV]HURNRSDVPRZ\PL
: ZLHOX ]DVWRVRZDQLDFK Z\PDJD VL EDUG]R GX*HM UH]\VWDQFML
G\QDPLF]QHM(UyGáDSUGXZV]HURNLP]DNUHVLH]PLDQQDSL FLD(UyGáD
2SLVDQH UR]ZL]DQLD XNáDGRZH (UyGHá SUGRZ\FK Z WHFKQRORJLL
ELSRODUQHM PRJ PLHü UyZQLH* ]DVWRVRZDQLH GR UHDOL]DFML (UyGHá
SUGRZ\FKZWHFKQRORJLL&026
1D U\V SU]HGVWDZLRQR VFKHPDW LGHRZ\ NDVNRGRZHJR OXVWUD
SUGRZHJRVWDQRZLFHRGSRZLHGQLNELSRODUQHJRXNáDGX]U\V
=DNáDGDMF*HWUDQ]\VWRU\0
0
VLGHQW\F]QH]HVFKHPDWXQD
U\VDRWU]\PXMHP\
U
U
U
U
U
U
DS
GS
GS
GS
GS
GS I
I
I
D
REF
O
1
2
2
1
2
=
−
=
=
=
=
U
U
DS
GS
1
=
:DUXQNLHPSUDF\WUDQ]\VWRUyZ0
L0
ZREV]DU]HQDV\FHQLDMHVWDE\
VSHáQLRQHE\á\QLHUyZQRFL
U
U
U
U
U
U
DS
GS
T
O
GS
T
2
2
2
≥
−
≥
−
,
5()
2
M
,
2
a)
3
M
4
M
1
M
8
*6
8
*6 2
8
*61
8
'6 2
8
'61
8
2
b)
8
*6
8
2
,
2
2
8
8
*6
7
−
(
)
2
8
8
*6
7
−
0 0 REV]DU OLQLRZ\
1
2
0 0 REV]DU QDV\FHQLD
1
2
0 REV]DU
QDV\FHQLD
0 REV]DU
OLQLRZ\
1
2
5\V .DVNRGRZH OXVWUR SUGRZH D VFKHPDW LGHRZ\ E FKDUDNWHU\VW\ND
SUGRZRQDSL FLRZD
:W\P]DNUHVLHQDSL üZ\MFLRZ\FKSUG(UyGáDMHVWVWDá\'DOV]H
]PQLHMV]DQLHQDSL FLD
U
O
SRZRGXMH]PQLHMV]HQLHQDSL FLD
U
DS 2
JG\*
QDSL FLH
U
DS1
MHVW XVWDORQH SU]H] NRáR QDSL ü
2
2
1
U
U
U
GS
GS
DS
−
=
LWUDQ]\VWRU 0
SUDFXMH GDOHM Z REV]DU]H QDV\FHQLD 7UDQ]\VWRU 0
]DF]\QDSUDFRZDüZREV]DU]HOLQLRZ\PDSUGZ\MFLRZ\
I
O
PDOHMH
FR MHVW VSRZRGRZDQH ]PQLHMV]DQLHP VL QDSL FLD
U
DS 2
: PRPHQFLH
JG\ ZDUWRü QDSL FLD
U
O
]PQLHMV]\ VL SRQL*HM
(
)
2 U
U
GS
T
−
RED
WUDQ]\VWRU\SUDFXMZREV]DU]HOLQLRZ\PU\VE
'\QDPLF]QUH]\VWDQFM Z\MFLRZOXVWUDNDVNRGRZHJRPR*HP\
Z\]QDF]\üQDSRGVWDZLHMHJRPDáRV\JQDáRZHJRVFKHPDWX]DVW SF]HJR
LMHJRNROHMQ\FKSU]HNV]WDáFHSU]HGVWDZLRQ\FKQDU\V>@
5H]\VWDQFMDWDZ\QRVL
(
)
[
]
(
)
(
)
r
U
I
U
r U
g
g
U
g
r
r
g r
r
g r
o
gs
ds
gs
m
ds
gs
ds
ds
ds
m
ds
ds
m
ds
=
=
−
+
+
−
=
+
+
≈
+
2
2
2
2
1
2
1
1
2
2
1
2
2
1
1
1
:VWRVXQNXGRSRGVWDZRZHJROXVWUDSUGRZHJR]U\VUH]\VWDQFMD
Z\MFLRZDOXVWUDNDVNRGRZHJRZ]URVáD
(
)
1
2
1
+
g r
m
ds
UD]\
3RQLHZD* SRWHQFMDá GUHQX WUDQ]\VWRUD 0
PR*H VL QLH]QDF]QLH
]PLHQLDü WR ]PLDQD WD SRZRGXMH ]PLDQ QDSL FLD (UyGáR SRGáR*H Z
WUDQ]\VWRU]H0
]DWHPZWUDQ]\VWRU]HW\PZ\VWSLHIHNWSRGáR*D
J
GV3
J 8
P
JV
3
3
J
GV4
J 8
P
JV
4
4
8
JV3
8
JV2
8
JV4
8
JV1
J 8
P
JV
2
2
J
GV2
J 8
P
JV
1
1
J
GV1
8
,
8
JV2
8
JV1
J 8
P
JV
2
2
J
GV2
J 8
P
JV
1
1
J
GV1
8
,
J
P3
J
GV3
J
GV4
J
P4
J 8
P
JV
2
2
J
GV2
J 8
GV
JV
1
2
8
,
J
GV1
(
)
J
J 8
P
GV
JV
2
1
2
+
8
JV2
5\V 0DáRV\JQDáRZ\ VFKHPDW ]DVW SF]\ OXVWUD NDVNRGRZHJR L MHJR NROHMQH
SU]HNV]WDáFHQLD
1D U\V SU]HGVWDZLRQR NRFRZ SRVWDü SU]HNV]WDáFRQHJR
VFKHPDWX ]DVW SF]HJR ] U\V Z NWyU\P XZ]JO GQLRQR VNáDGRZ
]PLHQQQDSL FLD(UyGáRSRGáR*H
U
bs2
WUDQ]\VWRUD0
J 8
PE
EV
2
2
J
GV2
8
,
J 8
P
JV
2
2
8
EV2
J
GV1
8
JV2
%2
5\V0DáRV\JQDáRZ\VFKHPDW
]DVW SF]\OXVWUDNDVNRGRZHJR
XZ]JO GQLDMF\HIHNWSRGáR*D
WUDQ]\VWRUD0
'\QDPLF]QD UH]\VWDQFMD Z\MFLRZD OXVWUD SU]\ XZ]JO GQLHQLX HIHNWX
SRGáR*DWUDQ]\VWRUD
(
)
[
]
r
r
r
g
g
r
o
ds
ds
m
mb
ds
=
+
+
+
1
2
2
2
1
1
MHVW QLHFR ZL NV]D QL* WD NWyU Z\]QDF]RQR SU]\ SRPLQL FLX HIHNWX
SRGáR*D
.DVNRGRZH OXVWUR SUGRZH ] WUDQ]\VWRUDPL 026)(7 ]DSHZQLD
]QDF]QLH ZL NV] G\QDPLF]Q UH]\VWDQFM Z\MFLRZ QL* MHJR
RGSRZLHGQLN ] WUDQ]\VWRUDPL ELSRODUQ\PL :DG NDVNRGRZHJR OXVWUD
SUGRZHJR]U\VMHVWVWRVXQNRZRGX*HQDSL FLHQDZHMFLXOXVWUD
UyZQH
2U
GS
RUD] VWRVXQNRZR GX*D ZDUWRü PLQLPDOQHJR QDSL FLD
Z\MFLRZHJRNWyUDQLHPR*H E\ü PQLHMV]D QL*
U
U
U
O
GS
T
min
≥
−
2
MDN
Z\MDQLRQR ZF]HQLHM SU]\ GDOV]\P REQL*DQLX QDSL FLD QD Z\MFLX
WUDQ]\VWRU 0
ZFKRG]L Z REV]DU OLQLRZ\ L SUG Z\MFLRZ\ PDOHMH
2JUDQLF]D WR ]QDF]QLH PR*OLZRü Z\NRU]\VWDQLD W\FK OXVWHU ZH
ZVSyáF]HVQ\FK DQDORJRZ\FK XNáDGDFK VFDORQ\FK &026 SUDFXM F\FK
SU]\QLVNLFKQDSL FLDFK]DVLODMF\FK
1D U\V SU]HGVWDZLRQR VFKHPDW LGHRZ\ QLVNRQDSL FLRZHJR
OXVWUDSUGRZHJRDQJKLJKVZLQJFXUUHQWPLUURUNWyUHZSRUyZQDQLX
GRNDVNRGRZHJROXVWUD]U\VSRVLDGDPQLHMV]HQDSL FLHZHMFLRZH
LZL NV]\]DNUHVQDSL FLDZ\MFLRZHJR>@
,
5()
2
M
,
2
3
M
4
M
1
M
8
*6
8
*6 2
8
*61
8
'62
8
'61
8
2
8
**
5\V 1LVNRQDSL FLRZH
OXVWUR SUGRZH W\SX ÄKLJK
VZLQJ´
=Dáy*P\GODXSURV]F]HQLDUR]ZD*D*HZV]\VWNLHWUDQ]\VWRU\0
0
PDM WH VDPH UR]PLDU\ L WH VDPH QDSL FLD
U
GS
1DSL FLH
U
GG
SRODU\]XMFH EUDPNL WUDQ]\VWRUyZ 0
L 0
QDOH*\ WDN GREUDü DE\
ZV]\VWNLHWUDQ]\VWRU\SUDFRZDá\ZREV]DUDFKQDV\FHQLD
=DWHP
U
U
U
U
U
U
U
U
DS
GS
T
GG
DS
GS
GS
T
1
1
2
min
min
≥
−
=
+
=
−
$E\WUDQ]\VWRU0
SUDFRZDáZREV]DU]HQDV\FHQLDSRWHQFMDáMHJRGUHQX
QLHPR*HE\üQL*V]\RG
U
U
GG
T
−
W]Q
8
8
*'
7
<
6WG QDSL FLH Z\MFLRZH
U
O
QLH PR*H REQL*\ü VL SRQL*HM SHZQHM
PLQLPDOQHMZDUWRFL
U
U
U
U
U
O
GG
T
GS
T
min
≥
−
=
−
2
2
3RUyZQXMF ]DOH*QRFL L ZLG]LP\ *H QDSL FLH
U
O min
OXVWUD SUGRZHJR W\SX ÄKLJK VZLQJ´ MHVW PQLHMV]H R ZDUWRü QDSL FLD
SURJRZHJR
U
T
RG QDSL FLD
U
O min
Z NRQZHQFMRQDOQ\P OXVWU]H
NDVNRGRZ\P]U\VD
%LRUF SRG XZDJ *H QDSL FLH SURJRZH
8
7
]DZLHUD VL Z JUDQLFDFK
±9 PD WR LVWRWQH ]QDF]HQLH SU]\ UHDOL]DFML QLVNRQDSL FLRZ\FK
DQDORJRZ\FKXNáDGyZ&026
5yZQLH* QDSL FLH ZHMFLRZH OXVWUD W\SX ÄKLJK VZLQJ´ MHVW PQLHMV]H
LZ\QRVL
U
GS
ZNODV\F]Q\POXVWU]HNDVNRGRZ\P
2U
GS
'UyGáDQDSL FLRZH
%DUG]RF] VWRSXQNW\SUDF\WUDQ]\VWRUyZZXQLSRODUQ\PXNáDG]LH
VFDORQ\P XVWDODQH V ]D SRPRF (UyGHá QDSL FLRZ\FK GRáF]RQ\FK GR
EUDPHNW\FKWUDQ]\VWRUyZ
:]DNUHVLH]PLHQQRSUGRZ\P EUDPNL W\FK WUDQ]\VWRUyZ ]ZDUWH V GR
PDV\ 'UyGáD QDSL FLRZH R WDNLP SU]H]QDF]HQLX PRJ SRVLDGDü GRü
GX*H UH]\VWDQFMH ZHZQ WU]QH SRQLHZD* REFL*RQH V EDUG]R GX*\PL
LPSHGDQFMDPL ZHMFLRZ\PL WUDQ]\VWRUyZ 026)(7 D SU]H] WR PRJ
E\üUHDOL]RZDQHZQDMSURVWV]HMSRVWDFLQSG]LHOQLNyZQDSL FLRZ\FK
6]F]HJyOQ\P URG]DMHP (UyGHá QDSL FLRZ\FK V XNáDG\
SU]HVXZDQLD QDSL FLD VWDáHJR NWyUH V VWRVRZDQH Z REZRGDFK
VSU] JDMF\FK GZyFK NROHMQ\FK VWRSQL XNáDGX VFDORQHJR RUD] QD
Z\MFLX RVWDWQLHJR VWRSQLD Z FHOX ]DSHZQLHQLD ]HURZHJR QDSL FLD
Z\MFLRZHJR SU]\ ]HURZ\P QDSL FLX ZHMFLRZ\P 3RQLHZD* XNáDG\
SU]HVXZDQLDQDSL FLD]QDMGXMVL ZJáyZQ\PWRU]HWUDQVPLVMLV\JQDáX
GODWHJR QLH SRZLQQ\ ZQRVLü WáXPLHQLD V\JQDáX X *\WHF]QHJR REFL*Dü
VWRSQLD SRSU]HGQLHJR RUD] SRZLQQ\ JZDUDQWRZDü VWHURZDQLH
QDSL FLRZHVWRSQLDQDVW SQHJR
'UyGáD QDSL FLRZH R PDáHM ZUD*OLZRFL QDSL FLD QD ]PLDQ\
WHPSHUDWXU\LQDSL FLD]DVLODQLDQD]\ZDVL (UyGáDPLRGQLHVLHQLD'UyGáD
QDSL FLRZH V Z ]QDF]QLH ZL NV]\P VWRSQLX ZUD*OLZH QD ]PLDQ\
WHPSHUDWXU\ QL* QD ]PLDQ\ QDSL FLD ]DVLODQLD RGZURWQLH QL* Z
SU]\SDGNX(UyGHáSUGRZ\FK
1D U\V SU]HGVWDZLRQR SU]\NáDG\ UHDOL]DFML G]LHOQLNyZ
QDSL FLRZ\FK Z WHFKQRORJLL 1026 L &026 ]EXGRZDQ\FK
]WUDQ]\VWRUyZ026)(7ZSRáF]HQLXGLRGRZ\P
3RQLHZD* SU]H] G]LHOQLN WUDQ]\VWRURZ\ SU]HSá\ZD WHQ VDP SU G
GUHQX
I
D
SU]\ SRPLMDOQLH PDá\FK SUGDFK REFL*H ]DWHP QDSL FLH
U
DSi
LWHJRWUDQ]\VWRUDZG]LHOQLNXMHVWUyZQH
(
)
U
I
K W L
U
DSi
D
i
T
=
+
'
1DSL FLHQDMW\PZ\MFLXG]LHOQLNDZ\QRVL
U
U
U
j
DSi
SS
i
j
=
−
=
∑
1
GOD M
N
SU]\F]\PNMHVWZRJyOQ\PSU]\SDGNXOLF]EWUDQ]\VWRUyZWZRU]F\FK
G]LHOQLN]DVLODQ\QDSL FLHPU
U
DD
SS
+
F]\OL
U
U
U
DD
SS
DSi
i
k
+
=
=
∑
1
2
M
,
'
1
M
8
'62
8
'61
8
3
M
8
'6
8
+
8
''
a)
2
M
,
'
1
M
8
'6 2
8
'61
8
3
M
8
'6
8
+
8
''
b)
8
'6
−
8
66
8
M
4
5\V']LHOQLNLQDSL FLRZH]EXGRZDQH]WUDQ]\VWRUyZ026)(7ZSRáF]HQLX
GLRGRZ\PDZWHFKQRORJLL1026EZWHFKQRORJLL&026
1DU\VSU]HGVWDZLRQRSURVWHSU]\NáDG\XNáDGyZSU]HVXZDQLD
QDSL FLDVWDáHJR
,
'
2
M
1
M
8
*6
a)
b)
,
'
X
2
X
,
8
*6
2
M
8
*6
,
X
,
X
2
−
8
66
+
8
''
5\V 8NáDG\ SU]HVXZDQLD QDSL FLD VWDáHJR D ]H (UyGáHP VWDáRSUGRZ\P E
ZWyUQLNRZ\
: XNáDG]LH QD U\VD SR]LRP QDSL FLD Z\MFLRZHJR
u
O
MHVW
SU]HVXQL W\Z]JO GHPSR]LRPXQDSL FLDZHMFLRZHJR
u
I
RVWDáZDUWRü
−
2U
GS
NWyU\SU]\LGHQW\F]Q\FKWUDQ]\VWRUDFK0
0
Z\QRVL
u
u
U
I
K W L
U
O
I
GS
D
T
− = −
= −
−
2
2
'
:XNáDG]LHQDU\VESU]HVXQL FLHWRZ\QRVL
−
U
GS
,QQH W\SRZH UR]ZL]DQLD VWRVRZDQH Z REZRGDFK ]DVLODQLD
XNáDGyZ VFDORQ\FK V SU]\WDF]DQH Z GDOV]\FK UR]G]LDáDFK SU]\
RPDZLDQLXNRQNUHWQ\FKUR]ZL]DXNáDGRZ\FK